ࡱ> DS^K|  bjbj ee/z TRRR#Rj%4>J>J>J>? KTlM,8i:i:i:i:i:i:i$lso$^iuN??NN^iJ>J>Hi:^:^:^NJ>J>8i:^N8i:^:^f__J>tW16Q ~_$ii0j_,p]*p__bpaNN:^NNNNN^i^i:^NNNjNNNNpNNNNNNNNN  : The Redesigned S A T Mathematics Sample Sets Information for users of assistive technology The document(s) that accompany these instructions are designed to be accessible to individuals who use screen readers, text readers, or other assistive technology. You may wish to consult the manual or help system of your application software to learn how best to take advantage of the following features implemented in this document. Headings Some questions include passages or other material that you may find it useful to return to or skip past. To assist in this kind of navigation, the test documents use headings as follows. Heading level 3: section titles Heading level 4: directions for a group of questions or references to material on which one or more questions will be based (for example, Question 3 is based on the following text:) Heading level 5: question numbers, which directly precede the associated questions Heading level 6: indications of skippable content (For example, you may prefer to skip some sections of this script, such as those that provide figure descriptions or possible answers in context for questions that involve revision. This content is identified at the beginning by the phrase Begin skippable content and at the end by the phrase End skippable content. These phrases are formatted as level6 headings.) Links This document includes hyperlinked material. There are two ways to follow a link. One is to move the flashing text cursor, or caret, into the hyperlinked text and press the Enter key; the other is to place the mouse cursor, or pointer, over the hyperlinked text and press Ctrl+leftclick (that is, press and release the left button on the mouse while holding down the Ctrl key on the keyboard). Some application software includes commands for listing links in a document. In JAWS, for example, pressing Insert+F5 provides a list of links. After following a link in Microsoft Word, you can return to your previous location by pressing Alt+leftarrow. Text attributes Boldfacing and underlining are used in this document for emphasis and in defined heading styles. Italic type is not used as an emphasis indicator in this document but is used in defined heading styles as well as where standard typographic conventions require them, such as book titles and mathematical variables. Adjust the settings of your screen reader or other application software if you wish to be notified of text attribute changes. Except where stated otherwise, this formatting is not critical to the meaning of the test material. Text and graphics size The styles used in this document result in text that is moderately enlarged. To enlarge text further in Microsoft Word, the following is recommended, in order of preference: 1.Adjust the styles to meet your needs. You can adjust both font size and typeface if desired. 2.Manually adjust the font size or typeface as desired. 3.Use Microsoft Words zoom function. This is the easiest way to enlarge any figures, but note that some screen readers will not read text that has moved off screen as a result of zooming. Pronunciation Some changes to the text have been made to improve the way screen readers pronounce the text where doing so would not inappropriately change test content. For example, we have inserted spaces between the letters of initialisms to ensure that the individual letters are spoken separately. However, please note that pronunciation errors may remain. If unsure of a word, use the spelling or characterbycharacter navigation function of your application software to resolve any uncertainties. Punctuation Where punctuation or symbols are critical to the meaning of test material, we either convert the symbol or punctuation mark to words (for example, it apostrophes or its apostrophe) or else include a statement advising you to take note of punctuation for a particular question or portion of a question. Tables Some questions may include tables. Use the tablenavigation features of your application software. Figures This document may include figures, which appear on screen. Following each figure on screen is text describing that figure. Readers using visual presentations of the figures may choose to skip parts of the text describing the figure that begin with Begin skippable figure description and end with End skippable figure description. Your application software may speak unhelpful information when you arrive at the figures, such as the figures size. If your application software offers a method of configuring speech for graphics, you may wish to use that to prevent it from speaking the unwanted information. Mathematical Equations and Expressions Some of these documents include mathematical equations and expressions. Some of the mathematical equations and expressions are presented as graphics. In cases where a mathematical equation or expression is presented as a graphic, a verbal presentation is also given and the verbal presentation comes directly after the graphic presentation. The verbal presentation is in green font to assist readers in telling the two presentation modes apart. Readers using audio alone can safely ignore the graphical presentations, and readers using visual presentations may ignore the verbal presentations. MathematicsCalculator The directions and question numbers below are representative of what students will encounter on test day. Some math sections allow the use of a calculator, while others do not, as indicated in the directions. Turn to Section 4 of your answer sheet to answer the questions in this section. For questions 1 through 30, solve each problem, choose the best answer from the choices provided, and fill in the corresponding circle on your answer sheet. For questions 31 through 38, solve the problem and enter your answer in the grid on the answer sheet. Please refer to the directions before question 31 on how to enter your answers in the grid. You may use any available space in your test booklet for scratch work. 1.The use of a calculator is permitted. 2.All variables and expressions used represent real numbers unless otherwise indicated. 3.Figures provided in this test are drawn to scale unless otherwise indicated. 4.All figures lie in a plane unless otherwise indicated. 5.Unless otherwise indicated, the domain of a given function f is the set of all real numbers x for which  EMBED Equation.DSMT4  f of x is a real number.  Begin skippable figure descriptions. The figure presents information for your reference in solving some of theproblems. Reference figure 1 is a circle with radius r. Two equations are presented below reference figure1. A equals pi times the square of r. C equals 2 pi r. Reference figure 2 is a rectangle with length ! and width w. An equation is presented below reference figure2. A equals !w. Reference figure 3 is a triangle with base b and height h. An equation is presented below reference figure3. A equals onehalfbh. Reference figure 4 is a right triangle. The two sides that form the right angle are labeled a and b, and the side opposite the right angle is labeled c. An equation is presented below reference figure4. c squared equals a squared plus b squared. Special Right Triangles Reference figure 5 is a right triangle with a 30degree angle and a 60degree angle. The side opposite the 30degree angle is labeled x. The side opposite the 60degree angle is labeledxtimes the squareroot of3. The side opposite the right angle is labeled2x. Reference figure 6 is a right triangle with two 45degree angles. Two sides are each labeleds. The side opposite the rightangle is labeled stimes the squareroot of2. Reference figure 7 is a rectangular solid whose base has length! and widthw and whose height ish. An equation is presented below reference figure7. V equals!wh. Reference figure 8 is a rightcircularcylinder whose base has radiusr and whose height ish. An equation is presented below reference figure8. V equalspitimes the square of rtimesh. Reference figure 9 is a sphere with radiusr. An equation is presented below reference figure9. V equalsfourthirds pi times the cube ofr. Reference figure 10 is a cone whose base has radiusr and whose height ish. Anequation is presented below reference figure10. V equals onethird times pi times the square of rtimesh. Reference figure 11 is an asymmetrical pyramid whose base has length! and widthw and whose height ish. An equation is presented below reference figure11. V equalsonethird!wh. Additional Reference Information The number of degrees of arc in a circle is360. The number of radians of arc in a circle is 2pi. The sum of the measures in degrees of the angles of a triangle is180. End skippable figure descriptions. For studentproduced response questions, students will also see the following directions: For questions 31 through 38, solve the problem and enter your answer in the grid, as described below, on the answer sheet. 1.Although not required, it is suggested that you write your answer in the boxes at the top of the columns to help you fill in the circles accurately. You will receive credit only if the circles are filled in correctly. 2.Mark no more than one circle in any column. 3.No question has a negative answer. 4.Some problems may have more than one correct answer. In such cases, grid only one answer. 5.Mixed numbers such as  EMBED Equation.DSMT4  three and onehalf must be recorded as  EMBED Equation.DSMT4  threepointfive or  EMBED Equation.DSMT4  sevenslashtwo. (If  EMBED Equation.DSMT4  three and onehalf is entered into the grid as , three,one,slash,two, it will be interpreted as  EMBED Equation.DSMT4  thirtyone halves, not  EMBED Equation.DSMT4  three and onehalf). 6.Decimal answers: If you obtain a decimal answer with more digits than the grid can accommodate, it may be either rounded or truncated, but it must fill the entire grid. The following are four examples of how to record your answer in the spaces provided. Keep in mind that there are four spaces provided to record each answer. Examples 1 and 2  Beging skippable figure description. Example 1: If your answer is a fraction such as seventwelfths, it should be recorded as follows. Enter seven in the first space, the fractionbar (aslash) in the second space, one in the third space, and two in the fourth space. All four spaces would be used in this example. Example 2: If your answer is a decimal value such as twopointfive, it could be recorded as follows. Enter two in the secondspace, the decimal point in the third space, and five in the fourth space. Only three spaces would be used in thisexample. End skippable figure description. Example 3  Beging skippable figure description. Example 3: Acceptable ways to record twothirds are: two slash three, point sixsixsix, and point sixsixseven. End skippable figure description. Example 4  Note: You may start your answers in any column, spacepermitting. Columns you dont need should be left blank. Beging skippable figure description. Example 4: It is not necessary to begin recording answers in the first space unless all four spaces are needed. For example, if your answer is 201, you may record two in the first space, zero in the second space, and one in the third space. Alternatively, you may record two in the second space, zero in the third space, and one in the fourth space. Spaces not needed should be left blank. End skippable figure description. Mathematics Sample Questions Question 1. The recommended daily calcium intake for a 20yearold is 1,000milligrams (mg). One cup of milk contains 299milligrams of calcium and one cup of juice contains 261milligrams of calcium. Which of the following inequalities represents the possible number of cups of milkm and cups of juicej a 20yearold could drink in a day to meet or exceed the recommended daily calcium intake from these drinks alone? A. EMBED Equation.DSMT4  299m plus 261j is greater than or equal to 1,000 B. EMBED Equation.DSMT4  299m plus 261j is greater than 1,000 C. EMBED Equation.DSMT4  the fraction 299 overm, plus the fraction 261 overj, is greater than or equal to1,000 D. EMBED Equation.DSMT4  the fraction 299 overm, plus the fraction 261 overj, is greater than1,000  HYPERLINK \l "_Explanation_for_question" Answer and Explanation. (Follow link to explanation of question1.) Question 2. A research assistant randomly selected 75undergraduate students from the list of all students enrolled in the psychologydegree program at a large university. She asked each of the 75students, How many minutes per day do you typically spend reading? The mean reading time in the sample was 89minutes, and the margin of error for this estimate was 4.28minutes. Another research assistant intends to replicate the survey and will attempt to get a smaller margin of error. Which of the following samples will most likely result in a smaller margin of error for the estimated mean time students in the psychologydegree program read per day? A.40 randomly selected undergraduate psychologydegree program students B.40 randomly selected undergraduate students from all degree programs at the college C.300 randomly selected undergraduate psychologydegree program students D.300 randomly selected undergraduate students from all degree programs at the college  HYPERLINK \l "_Explanation_for_question_28" Answer and Explanation. (Follow link to explanation of question2.) Questions 3 through 5 refer to the following information and figure. The first metacarpal bone is located in the wrist. The following scatterplot shows the relationship between the length of the first metacarpal bone and height for 9people. The line of best fit is also shown.  Begin skippable figure description. The figure presents a gridded graph titled Height of Nine People and Length of Their First Metacarpal Bone and nine data points. The yaxis is labeled Length of first metacarpal bone, in centimeters, and the xaxis is labeled Height, in centimeters. The values 4, 4.5, and 5 are labeled on the xaxis with a vertical grid line at every increment of 0.1. The values 155 through 185, in increments of 5, are labeled on the yaxis with a horizontal grid line at every increment ofone. The approximate values of the nine data points on the scatterplot are as follows. 4.0 comma 157. 4.1 comma 163. 4.3 comma 175. 4.5 comma 171. 4.6 comma 173. 4.7 comma 173. 4.8 comma 172. 4.9 comma 183. 5.0 comma178. A straight line of best fit is drawn for the data points. The approximate coordinates of the line are as follows. 4.0 comma 161.5 4.1 comma 163. 4.2 comma 165. 4.3 comma 167. 4.4 comma 169. 4.5 comma 171. 4.6 comma 172.5. 4.7 comma 174.5. 4.8 comma 176. 4.9 comma 178. 5.0 comma 180. End skippable figure description. Question 3 refers to the  HYPERLINK \l "_Questions_3_through" information and figure (followlink) provided on pages15through16. Question 3. How many of the nine people have an actual height that differs by more than 3centimeters from the height predicted by the line of best fit? A.2 B.4 C.6 D.9  HYPERLINK \l "_Explanation_for_question_1" Answer and Explanation. (Follow link to explanation of question3.) Question 4 refers to the  HYPERLINK \l "_Questions_3_through" information and figure (follow link) provided on pages15through16. Question 4. Which of the following is the best interpretation of the slope of the line of best fit in the context of this problem? A.The predicted height increase in centimeters for one centimeter increase in thefirst metacarpal bone B.The predicted first metacarpal bone increase in centimeters for every centimeterincrease in height C.The predicted height in centimeters of a person with a first metacarpal bone length of 0centimeters D.The predicted first metacarpal bone length in centimeters for a person with a height of 0centimeters  HYPERLINK \l "_Explanation_for_question_2" Answer and Explanation. (Follow link to explanation of question4.) Question 5 refers to the  HYPERLINK \l "_Questions_3_through" information and figure (follow link) provided on pages15through16. Question 5. Based on the line of best fit, what is the predicted height for someone with a first metacarpal bone that has a length of 4.45 centimeters? A.168 centimeters B.169 centimeters C.170 centimeters D.171 centimeters  HYPERLINK \l "_Explanation_for_question_3" Answer and Explanation. (Follow link to explanation of question5.) Question 6. Aaron is staying at a hotel that charges $99.95 per night plus tax for a room. A tax of 8% is applied to the room rate, and an additional onetime untaxed fee of $5.00 is charged by the hotel. Which of the following represents Aarons total charge, in dollars, for staying xnights? A. EMBED Equation.DSMT4  parenthesis, 99.95 plus 0.08x, close parenthesis, plus5 B. EMBED Equation.DSMT4  1.08, parenthesis, 99.95x, close parenthesis, plus5 C. EMBED Equation.DSMT4  1.08, parenthesis, 99.95x plus5, closeparenthesis D. EMBED Equation.DSMT4  1.08, parenthesis, 99.95 plus5, closeparenthesis,x  HYPERLINK \l "_Explanation_for_question_4" Answer and Explanation. (Follow link to explanation of question6.) Question 7 refers to the following figure and system of threeequations.  Begin skippable figure description. The figure presents the graph of a circle, a parabola, and a line in the xyplane. Thehorizontal axis is labeled x, the vertical axis is labeled y, and the origin is labeledO. The integers negative3 through 3 appear on both axes. The circle has its center at the origin and radius of approximately2.2. The parabola has its vertex on the yaxis at negative3 and opens upward. The circle and parabola intersect at four points, of which two are below the xaxis and two are above the xaxis. Of the two points of intersection below the xaxis, one is to the left of the yaxis and one is to the right of the yaxis. Of the two points of intersection above the xaxis, one is to the left of the yaxis and one is to the right of the yaxis. The line slants upward and to the right, and passes through two of the four points of intersection where the circle and parabola meet, one below the xaxis and to the left of the yaxis, and one above the xaxis to the right of the yaxis. In other words, the three graphs intersect at two points. End skippable figure description. The following system of three equations is given beneath the figure.  EMBED Equation.DSMT4  xsquared plus ysquared equalsfive.  EMBED Equation.DSMT4  y equals x squared minusthree.  EMBED Equation.DSMT4  x minusy equalsone. Question 7. A system of three equations and their graphs in the xyplane are shown above. How many solutions does the system have? A.One B.Two C.Three D.Four  HYPERLINK \l "_Explanation_for_question_5" Answer and Explanation. (Follow link to explanation of question7.) Question 8 refers to the following information and table. The following table classifies 103 elements as metal, metalloid, or nonmetal and as solid, liquid, or gas at standard temperature and pressure. SolidsLiquidsGasesTotalMetals771078Metalloids7007Nonmetals611118Total90211103 Question 8. What fraction of all solids and liquids in the preceding table are metalloids?  HYPERLINK \l "_Explanation_for_question_6" Answer and Explanation. (Follow link to explanation of question8.) Question 9. If  EMBED Equation.DSMT4  the fraction negative9 over5 is less than negative3t plus1, and negative3t plus1 is less than the negative of the fraction 7over4, what is one possible value of  EMBED Equation.DSMT4  9t minus3?  HYPERLINK \l "_Explanation_for_question_7" Answer and Explanation. (Follow link to explanation of question9.) Questions 10 and 11 refer to the following information andtable. A survey was conducted among a randomly chosen sample of U.S.citizens about U.S. voter participation in the November2012 presidential election. The following table displays a summary of the survey results. Reported Voting by Age (in thousands) VotedDid Not VoteNo ResponseTotal18 to 34yearolds30,32923,2119,46863,00835 to 54yearolds47,08517,7219,47674,28255 to 74yearolds43,07510,0926,83159,998People 75 years old and over12,4593,5081,82717,794Total132,94854,53227,602215,082 Question 10 refers to the preceding  HYPERLINK \l "_Questions_10_and" information and table (follow link). Question 10. According to  HYPERLINK \l "Title_2" the table (followlink), for which age group did the greatest percentage of people report that they had voted? A.18- to 34-year-olds B.35- to 54-year-olds C.55- to 74-year-olds D.People 75 years old and over  HYPERLINK \l "_Explanation_for_question_8" Answer and Explanation. (Follow link to explanation of question10.) Question 11 refers to the  HYPERLINK \l "_Questions_10_and" information and table (follow link) provided on page24. Question 11. Of the 18- to 34yearolds who reported voting, 500people were selected at random to do a followup survey where they were asked which candidate they voted for. There were 287people in this followup survey sample who said they voted for CandidateA, and the other 213people voted for someone else. Using the data from both the followup survey and the initial survey, which of the following is most likely to be an accurate statement? A.About 123millionpeople 18 to 34 yearsold would report voting for CandidateA in the November2012 presidentialelection. B.About 76millionpeople 18 to 34 yearsold would report voting for CandidateA in the November2012 presidentialelection. C.About 36millionpeople 18 to 34 yearsold would report voting for CandidateA in the November2012 presidentialelection. D.About 17millionpeople 18 to 34 yearsold would report voting for CandidateA in the November2012 presidentialelection.  HYPERLINK \l "_Explanation_for_question_9" Answer and Explanation. (Follow link to explanation of question11.) Question 12. A companys manager estimated that the cost C, in dollars, of producing n items is  EMBED Equation.DSMT4  Cequals 7n plus350. The company sells each item for $12. The company makes a profit when total income from selling a quantity of items is greater than the total cost of producing that quantity of items. Which of the following inequalities gives all possible values of n for which the manager estimates that the company will make aprofit? A. EMBED Equation.DSMT4 n is less than 70. B. EMBED Equation.DSMT4 n is less than 84. C. EMBED Equation.DSMT4 n is greater than 70. D. EMBED Equation.DSMT4 n is greater than 84.  HYPERLINK \l "_Explanation_for_question_29" Answer and Explanation. (Follow link to explanation of question12.) Question 13. At a primate reserve, the mean age of all the male primates is 15years, and the mean age of all female primates is 19years. Which of the following must be true about the mean agem of the combined group of male and female primates at the primatereserve? A. EMBED Equation.DSMT4  m equals 17. B. EMBED Equation.DSMT4  m is greater than 17. C. EMBED Equation.DSMT4  m is less than 17. D. EMBED Equation.DSMT4  15 is less than m, and m is less than 19.  HYPERLINK \l "_Explanation_for_question_10" Answer and Explanation. (Follow link to explanation of question13.) Question 14. A researcher wanted to know if there is an association between exercise and sleep for the population of 16yearolds in the UnitedStates. She obtained survey responses from a random sample of 2000 UnitedStates 16yearolds and found convincing evidence of a positive association between exercise and sleep. Which of the following conclusions is wellsupported by the data? A.There is a positive association between exercise and sleep for 16yearolds in the UnitedStates. B.There is a positive association between exercise and sleep for 16yearolds in the world. C.Using exercise and sleep as defined by the study, an increase in sleep is caused by an increase of exercise for 16yearolds in the UnitedStates. D.Using exercise and sleep as defined by the study, an increase in sleep is caused by an increase of exercise for 16yearolds in theworld.  HYPERLINK \l "_Explanation_for_question_11" Answer and Explanation. (Follow link to explanation of question14.) Question 15. A biology class at Central High School predicted that a local population of animals will double in size every 12years. The population at the beginning of 2014 was estimated to be 50animals. If P represents the population nyears after 2014, then which of the following equations represents the classs model of the population over time? A. EMBED Equation.DSMT4  P equals 12 plus 50n. B. EMBED Equation.DSMT4  P equals 50 plus 12n. C. EMBED Equation.DSMT4  P equals 50, parenthesis, 2, close parenthesis, to the power of 12n. D. EMBED Equation.DSMT4  P equals 50, parenthesis, 2, close parenthesis, to the power of the fraction nover12.  HYPERLINK \l "_Explanation_for_question_12" Answer and Explanation. (Follow link to explanation of question15.) Question 16 is based on the following figure.  Begin skippable figure description. The figure presents line segments AE and BD that intersect at pointC. Line segments AB and DE are drawn resulting in two triangles ABC and EDC. A note under the figure says that the figure is not drawn to scale. End skippable figure description. Question 16. In the  HYPERLINK \l "_Question_16_is" preceding figure(followlink),  EMBED Equation.DSMT4 triangle ABC is similar to triangle EDC. Which of the following must be true? A. EMBED Equation.DSMT4 Line segment AE is parallel to line segment BD. B. EMBED Equation.DSMT4 Line segment AE is perpendicular to line segment BD. C. EMBED Equation.DSMT4 Line segment AB is parallel to line segment DE. D. EMBED Equation.DSMT4 Line segment AB is perpendicular to line segment DE.  HYPERLINK \l "_Explanation_for_question_13" Answer and Explanation. (Follow link to explanation of question16.) Question 17. The gas mileage for Peters car is 21miles per gallon when the car travels at an average speed of 50milesperhour. The cars gas tank has 17gallons of gas at the beginning of a trip. If Peters car travels at an average speed of 50milesperhour, which of the following functions fmodels the number of gallons of gas remaining in the tankthours after the trip begins? A. EMBED Equation.DSMT4  f oft equals 17 minus the fraction 21 over 50t. B. EMBED Equation.DSMT4  f oft equals 17 minus the fraction 50t over 21. C. EMBED Equation.DSMT4  f oft equals the fraction whose numerator is 17 minus 21t, and whose denominator is 50. D. EMBED Equation.DSMT4  f oft equals the fraction whose numerator is 17 minus 50t, and whose denominator is 21.  HYPERLINK \l "_Explanation_for_question_14" Answer and Explanation. (Follow link to explanation of question17.) Question 18. The toll rates for crossing a bridge are $6.50 for a car and $10 for a truck. During a twohour period, a total of 187 cars and trucks crossed the bridge, and the total collected in tolls was $1,338. Solving which of the following systems of equations yields the number of cars,x, and the number of trucks,y, that crossed the bridge during the twohours? A. EMBED Equation.DSMT4 x plus y equals 1,338.  EMBED Equation.DSMT4 6.5x plus 10y equals187. B. EMBED Equation.DSMT4  x plus y equals 187.  EMBED Equation.DSMT4  6.5x, plus 10y, equals the fraction 1,338 over2. C. EMBED Equation.DSMT4  x plus y equals 187.  EMBED Equation.DSMT4  6.5x, plus 10y equals 1,338. D. EMBED Equation.DSMT4  x plus y equals 187.  EMBED Equation.DSMT4 6.5x, plus 10y equals 1,338 times2.  HYPERLINK \l "_Explanation_for_question_15" Answer and Explanation. (Follow link to explanation of question18.) Question 19. When a scientist dives in salt water to a depth of 9feet below the surface, thepressure due to the atmosphere and surrounding water is 18.7poundspersquareinch. As the scientist descends, the pressure increases linearly. At a depth of 14feet, the pressure is 20.9poundspersquareinch. If the pressure increases at a constant rate as the scientists depth below the surface increases, which of the following linear models best describes the pressurep in poundspersquareinch at a depth of dfeet below the surface? A. EMBED Equation.DSMT4  pequals 0.44d plus 0.77. B. EMBED Equation.DSMT4  pequals 0.44d plus 14.74. C. EMBED Equation.DSMT4  pequals 2.2d minus 1.1. D. EMBED Equation.DSMT4  pequals 2.2d minus 9.9.  HYPERLINK \l "_Explanation_for_question_16" Answer and Explanation. (Follow link to explanation of question19.) Question 20 is based on the following figure.  Begin skippable figure description. The figure, titled Count of Manatees, presents the graph of a scatterplot with a line. The horizontal axis is labeled Year, and the vertical axis is labeled Number of Manatees. The years 1990 through 2015 are labeled on the horizontal axis, in increments of 5 years. The numbers 1,000 through 6,000 are labeled on the vertical axis, in increments of 1,000. Grid lines extend from the labeled increments of both axes. There are 24 data points on the graph. The data points range horizontally from years 1991 to 2011 and vertically from approximately 1,300 manatees to approximately 5,100 manatees. A line of best fit is drawn for the range of years represented by the data points. The line begins at year 1991 with approximately 1,200 manatees and ends at year 2011 with approximately 4,200 manatees. The line of best fit intersects four vertical grid lines, which represent 5year increments, at the following approximate values. Year 1995: 1,800 manatees. Year 2000: 2,600 manatees. Year 2005: 3,300 manatees. Year 2010: 4,100 manatees. End skippable figure description. Question 20. The preceding  HYPERLINK \l "_Question_20_is" scatterplot (follow link) shows counts of Florida manatees, a type of sea mammal, from 1991 to 2011. Based on the line of best fit to the data shown, whichof the following values is closest to the average yearly increase in the number of manatees? A.0.75 B.75 C.150 D.750  HYPERLINK \l "_Explanation_for_question_17" Answer and Explanation. (Follow link to explanation of question20.) Question 21 is based on the following figure.  Begin skippable figure description. The figure, titled Bacteria Growth, presents a graph of two curved lines. The horizontal axis is labeled Time in hours and the vertical axis is labeled Area covered in square centimeters. Both axes are labeled from 0 to 10 in increments of one with grid lines extending from each labeled increment. The curved line labeled Dish1 begins on the vertical axis at 1 and curves steeply up and to the right passing through the point with coordinates  EMBED Equation.DSMT4  2comma4, and the point with coordinates  EMBED Equation.DSMT4  3comma8. The curved line labeled Dish2 begins on the vertical axis at 2 and moves to the right before curving gradually up and to the right passing through the point with coordinates  EMBED Equation.DSMT4  3 comma 3, and the point with coordinates  EMBED Equation.DSMT4  5 comma 6. The two curved lines intersect at a point with approximate coordinates  EMBED Equation.DSMT4  1.2 comma 2.1. End skippable figure description. Question 21. A researcher places two colonies of bacteria into two petri dishes that each have area 10squarecentimeters. After the initial placement of the bacteria  EMBED Equation.DSMT4  parenthesis, tequalszero, close parenthesis, the researcher measures and records the area covered by the bacteria in each dish every tenminutes. The data for each dish were fit by a smooth curve, as shown in  HYPERLINK \l "_Question_21_is" the figure (follow link), where each curve represents the area of a dish covered by bacteria as a function of time, inhours. Which of the following is a correct statement about the precedingdata? A.At time  EMBED Equation.DSMT4  tequals zero, both dishes are 100% covered by bacteria. B.At time  EMBED Equation.DSMT4  tequals zero, bacteria covers 10% of Dish1 and 20% of Dish2. C.At time  EMBED Equation.DSMT4  tequals zero, Dish2 is covered with 50% more bacteria thanDish1. D.For the first hour, the area covered in Dish2 is increasing at a higher average rate than the area covered in Dish1.  HYPERLINK \l "_Explanation_for_question_18" Answer and Explanation. (Follow link to explanation of question21.) Question 22. A typical image taken of the surface of Mars by a camera is 11.2gigabits in size. A tracking station on Earth can receive data from the spacecraft at a data rate of 3megabitspersecond for a maximum of 11hours each day. If 1gigabit equals 1,024megabits, what is the maximum number of typical images that the tracking station could receive from the camera eachday? A.3 B.10 C.56 D.144  HYPERLINK \l "_Explanation_for_question_19" Answer and Explanation. (Follow link to explanation of question22.) Question 23.  EMBED Equation.DSMT4  x squared plus y squared equals 153.  EMBED Equation.DSMT4  y equals negative 4x. If  EMBED Equation.DSMT4  parenthesis, x comma y, close parenthesis, is a solution to the preceding system of equations, what is the value of  EMBED Equation.DSMT4  xsquared? A. EMBED Equation.DSMT4  negative 51 B.3 C.9 D.144  HYPERLINK \l "_Explanation_for_question_20" Answer and Explanation. (Follow link to explanation of question23.) Question 24 is based on the following figure.  Begin skippable figure description. The figure presents a metal nut with two hexagonal faces and six sides. The thickness of one side, from one hexagonal face to the other hexagonal face of the nut, is labeled as 1centimeter. End skippable figure description. Question 24. The preceding  HYPERLINK \l "_Question_24_is" figure (follow link) shows a metal hex nut with two regular hexagonal faces and a thickness of 1centimeter. The length of each side of a hexagonal face is 2centimeters. A hole with a diameter of 2centimeters is drilled through the nut. The density of the metal is 7.9grams per cubiccentimeter. What is the mass of this nut, to the nearest gram? (Density is mass divided by volume.)  HYPERLINK \l "_Explanation_for_question_21" Answer and Explanation. (Follow link to explanation of question24.) Questions 25 and 26 refer to the following information. An international bank issues its Traveler credit cards worldwide. When a customer makes a purchase using a Traveler card in a currency different from the customers home currency, the bank converts the purchase price at the daily foreign exchange rate and then charges a 4% fee on the converted cost. Sara lives in the UnitedStates, but is on vacation in India. She used her Traveler card for a purchase that cost 602rupees (Indiancurrency). The bank posted a charge of $9.88 to her account that included the 4% fee. Question 25 refers to the  HYPERLINK \l "_Questions_25_and" information (follow link) provided at the top of page40. Question 25. What foreign exchange rate, in Indian rupees per one U.S.dollar, did thebank use for Saras charge? Round your answer to the nearest whole number.  HYPERLINK \l "_Explanation_for_question_23" Answer and Explanation. (Follow link to explanation of question25.) Question 26 refers to the  HYPERLINK \l "_Questions_25_and" information (follow link) provided at the top of page40. Question 26. A bank in India sells a prepaid credit card worth 7,500rupees. Sara canbuy the prepaid card using dollars at the daily exchange rate with no fee,but she will lose any money left unspent on the prepaid card. What is theleast number of the 7,500rupees on the prepaid card Sara must spendfor the prepaid card to be cheaper than charging all her purchases onthe Traveler card? Round your answer to the nearest whole number ofrupees.  HYPERLINK \l "_Explanation_for_question_22" Answer and Explanation. (Follow link to explanation of question26.) Question 27. If k is a positive constant different from 1, which of the following could be the graph of  EMBED Equation.DSMT4  y minus x equals, k, parenthesis, x plus y, close parenthesis, in the xyplane? Each of the four answer choices presents a graph in the xyplane. The numbers  EMBED Equation.DSMT4  negative6 through 6 appear along both axes, and the origin is labeledO. A. Begin skippable figure description. Choice A. The graph shows a line that goes up from left to right and intersects the xaxis at negative2, and the yaxis at2. End skippable figure description. B. Begin skippable figure description. Choice B. The graph shows a line that goes down from left to right and goes through the origin. The line appears to pass through the point with coordinates negative1 comma2. End skippable figure description. C. Begin skippable figure description. Choice C. The graph shows a line that goes up from left to right and intersects the yaxis at negative3, and the xaxis at 1.5. End skippable figure description. D. Begin skippable figure description. Choice D. The graph shows a smooth curve that appears to be a parabola. The parabola has its vertex at the origin, opens upward, and passes through the point with coordinates 1comma1. End skippable figure description.  HYPERLINK \l "_Explanation_for_question_24" Answer and Explanation. (Follow link to explanation of question27.) Question 28. The function f is defined by  EMBED Equation.DSMT4  fofx equals 2xcubed, plus 3xsquared, plus cx, plus8, where c is a constant. In the xyplane, the graph off intersects the xaxis at the three points  EMBED Equation.DSMT4  parenthesis, negative4 comma0, close parenthesis,  EMBED Equation.DSMT4  parenthesis, onehalf comma0, close parenthesis, and  EMBED Equation.DSMT4  parenthesis, p comma0, close parenthesis. What is the valueofc? A. EMBED Equation.DSMT4  negative 18 B. EMBED Equation.DSMT4  negative 2 C.2 D.10  HYPERLINK \l "_Explanation_for_question_25" Answer and Explanation. (Follow link to explanation of question28.) Question 29. If the expression EMBED Equation.DSMT4  the fraction whose numerator is 4xsquared, and whose denominator is 2xminusone, is written in the equivalent form EMBED Equation.DSMT4 thefraction whose numerator is one, and whose denominator is 2xminusone, plus A, what is A in terms ofx? A. EMBED Equation.DSMT4  2 x plus one. B. EMBED Equation.DSMT4  2 x minus one. C. EMBED Equation.DSMT4  4 x squared. D. EMBED Equation.DSMT4  4 x squared minusone.  HYPERLINK \l "_Explanation_for_question_26" Answer and Explanation. (Follow link to explanation of question29.) Question 30 is based on the following information and figure. An architect drew the following sketch while designing a house roof. The dimensions shown are for the interior of the triangle.  Begin skippable figure description. The figure presents a triangle with a horizontal base. Labels are given to two sides and to two angles. The left side of the triangle is labeled 24feet. The base of the triangle is labeled 32feet. The two lower interior angles are labeled xdegrees. A note under the figure says that the figure is not drawn to scale. End skippable figure description. Question 30. What is the value of  EMBED Equation.DSMT4 cosine x?  HYPERLINK \l "_Explanation_for_question_27" Answer and Explanation. (Follow link to explanation of question30.) Mathematics Sample Question Answers and Explanations The following are explanations of answers to sample questions 1 through30. The heading of each explanation is hyperlinked to the actual question. In addition, each explanation is followed by two hyperlinks: one to the question explained and one to the next question. There are two ways to follow a link. One is to move the flashing text cursor, or caret, into the hyperlinked text and press the Enter key; the other is to place the mouse cursor, or pointer, over the hyperlinked text and press Ctrl+leftclick (that is, press and release the left button on the mouse while holding down the Ctrl key on the keyboard). After following a link in Microsoft Word, you can return to your previous location (for example, the answer explanation) by pressing Alt+leftarrow. Explanation for  HYPERLINK \l "_Question_1." question 1. (Follow link back to original question.) Program: SAT, PSAT/NMSQT, PSAT 10 Subscore: Heart of algebra Calculator usage: Yes Objective: Students must identify the correct mathematical notation for an inequality to represent a realworld situation. Difficulty: Easy Key: A Choice A is correct. Multiplying the number of cups of milk by the amount of calcium each cup contains and multiplying the number of cups of juice by the amount of calcium each cup contains gives the total amount of calcium from each source. The student must then find the sum of these two numbers to find the total amount of calcium for the day. Because the question asks for the calcium from these two sources to meet or exceed the recommended daily intake, the sum of these two products must be greater than or equal to 1,000. Choice B is not the correct answer. This answer may result from a misunderstanding of the meaning of inequality symbols as they relate to reallife situations. This answer does not allow for the daily intake to meet the recommended daily amount. Choice C is not the correct answer. This answer may result from a misunderstanding of proportional relationships. Here the wrong operation is applied, with the total amount of calcium per cup divided by the number of cups of each type of drink. These values should be multiplied. Choice D is not the correct answer. This answer may result from a combination of mistakes. The inequality symbol used allows the option to exceed, but not to meet, the recommended daily value, and the wrong operation may have been applied when calculating the total amount of calcium intake from each drink. Link back to  HYPERLINK \l "_Question_1." question 1. Link back to  HYPERLINK \l "_Question_2." question 2. Explanation for  HYPERLINK \l "_Question_2." question 2. (Follow link back to original question.) Program: SAT Subscore: Problem solving and data analysis Calculator usage: Yes Objective: Students must first read and understand statistics calculated from a survey that was conducted. Then, students must use apply their knowledge about the relationship between sample size and subject selection on margin of error. Difficulty: Easy Key: C Choice C is correct. Increasing the sample size while randomly selecting participants from the original population of interest will most likely result in a decrease in the margin of error. Choice A is not the correct answer. This answer may result from a misunderstanding of the importance of sample size to a margin of error. The margin of error is likely to increase with a smaller sample size. Choice B is not the correct answer. This answer may result from a misunderstanding of the importance of sample size and participant selection to a margin of error. The margin of error is likely to increase due to the smaller sample size. Also undergraduate students from all degree programs at the college is a different population than the original survey and therefore the impact to the mean and margin of error cannot be predicted. Choice D is not the correct answer. This answer may result from a misunderstanding of participant selection to a margin of error. Undergraduate students from all degree programs at the college is a different population than the original survey and therefore the impact to the mean and margin of error cannot be predicted. Link back to  HYPERLINK \l "_Question_2." question 2. Link back to  HYPERLINK \l "_Question_3." question 3. Explanation for  HYPERLINK \l "_Question_3." question 3. (Follow link back to original question.) Program: SAT, PSAT/NMSQT, PSAT 10 Subscore: Problem solving and data analysis Calculator usage: Yes Objective: Students must read and interpret information from a data display. Difficulty: Easy Key: B Choice B is correct. The people who have first metacarpal bones of length 4.0, 4.3, 4.8, and 4.9centimeters have heights that are greater than 3centimeters different from the height predicted by the line of best fit. Choice A is not the correct answer. There are 2people whose actual heights are more than 3centimeters above the height predicted by the line of best fit. However, there are also 2people whose actual heights are farther than 3centimeters below the line of best fit. Choice C is not the correct answer. There are 6data points in which the absolute value of the between the actual height and the height predicted by the line of best fit is greater than 1centimeter. Choice D is not the correct answer. The data on the graph represents 9different people; however, the absolute value of the difference between actual height and predicted height is not greater than 3for all of the people. Link back to  HYPERLINK \l "_Question_3." question 3. Link back to  HYPERLINK \l "_Question_4." question 4. Explanation for  HYPERLINK \l "_Question_4." question 4. (Follow link back to original question.) Program: SAT, PSAT/NMSQT, PSAT 10 Subscore: Heart of algebra Calculator usage: Yes Objective: Students must interpret the meaning of the slope of the line of best fit in the context provided. Difficulty: Easy Key: A Choice A is correct. The slope is the change in the vertical distance divided by the change in the horizontal distance between any two points on a line. In this context, the change in the vertical distance is the change in the predicted height of a person, and the change in the horizontal distance is the change in the length of his or her first metacarpal bone. The unit rate, or slope, is the increase in predicted height for each increase of 1centimeter of the first metacarpalbone. Choice B is not the correct answer. Students who select this answer may have interpreted slope incorrectly as run over rise. Choice C is not the correct answer. Students who select this answer may have mistaken slope for the yintercept. Choice D is not the correct answer. Students who select this answer may have mistaken slope for the xintercept. Link back to  HYPERLINK \l "_Question_4." question 4. Link back to  HYPERLINK \l "_Question_5." question 5. Explanation for  HYPERLINK \l "_Question_5." question 5. (Follow link back to original question.) Program: SAT, PSAT/NMSQT, PSAT 10 Subscore: Problem solving and data analysis Calculator usage: Yes Objective: Students must use the line of best fit to make a prediction. Students must also demonstrate fluency in reading graphs and decimal numbers. Difficulty: Easy Key: C Choice C is correct. Students must see that the scale of the xaxis is 0.1, and therefore the xvalue of 4.45 is halfway between the unmarked value of 4.4 and the marked value of 4.5. The student must then find the yvalue on the line of best fit that corresponds with an xvalue of 4.45, which is170. Choice A is not the correct answer. A student who mistakenly finds the point on the line between the xvalues of 4.3 and 4.4 will find a predicted metacarpal bone length of 168centimeters. Choice B is not the correct answer. A student who mistakenly finds the point on the line that corresponds with an xvalue of 4.4centimeters will find a predicted height of approximately 169centimeters. Choice D is not the correct answer. A student who mistakenly finds the point on the line that corresponds with an xvalue of 4.5centimeters will find a predicted height of approximately 171centimeters. Students might also choose this option if they mistakenly use the data point which has an xvalue closest to 4.45centimeters. Link back to  HYPERLINK \l "_Question_5." question 5. Link back to  HYPERLINK \l "_Question_6." question 6. Explanation for  HYPERLINK \l "_Question_6." question 6. (Follow link back to original question.) Program: SAT, PSAT/NMSQT, PSAT 10 Subscore: Heart of algebra Calculator usage: Yes Objective: Students must create a linear expression to represent a realworld situation. Difficulty: Easy Key: B Choice B is correct. The total charge that Aaron will pay is the room rate, the 8% tax on the room rate, and a fixed fee. If Aaron stayed xnights, then the total charge is  EMBED Equation.DSMT4  parenthesis, 99.95x plus 0.08, times 99.95x, close parenthesis, plus5, which can be rewritten as EMBED Equation.DSMT4  1.08, parenthesis, 99.95x, close parenthesis, plus5. Choice A is not the correct answer. The expression includes only one night stay in the room and does not accurately account for tax on the room. Choice C is not the correct answer. The expression includes tax on the fee, and the hotel does not charge tax on the $5.00fee. Choice D is not the correct answer. The expression includes tax on the fee and a fee charge for each night. Link back to  HYPERLINK \l "_Question_6." question 6. Link back to  HYPERLINK \l "_Question_7." question 7. Explanation for  HYPERLINK \l "_Question_7." question 7. (Follow link back to original question.) Program: SAT Subscore: Passport to advanced math Calculator usage: Yes Objective: Students must demonstrate their conceptual understanding of graphical representations of variable relationships. Difficulty: Easy Key: B Choice B is correct. The solutions to the system of equations are the points where the circle, parabola, and line all intersect. These points are  EMBED Equation.DSMT4  parenthesis, negative one comma two, close parenthesis, and  EMBED Equation.DSMT4  parenthesis, twocommaone, close parenthesis, and these are the only solutions to the system. Choice A is not the correct answer. This answer may reflect the misconception that a system of equations can have only one solution. Choice C is not the correct answer. This answer may reflect the misconception that a system of equations has as many solutions as the number of equations in the system. Choice D is not the correct answer. This answer may reflect the misconception that the solutions of the system are represented by the points where any two of the curves intersect, rather than the correct concept that the solutions are represented only by the points where all three curves intersect. Link back to  HYPERLINK \l "_Question_7." question 7. Link back to  HYPERLINK \l "_Question_8." question 8. Explanation for  HYPERLINK \l "_Question_8." question 8. (Follow link back to original question.) Program: SAT, PSAT/NMSQT, PSAT 10 Subscore: Problem solving and data analysis Calculator usage: Yes Objective: Students must read information from two-way tables and determine the specific relationship between two categorical variables. Difficulty: Easy Key:  EMBED Equation.DSMT4  the fraction 7 over 92 There are 7 metalloids that are solid or liquid, and there are 92 total solids and liquids. Therefore, the fraction of solids and liquids that are metalloids is  EMBED Equation.DSMT4  thefraction 7over92. Link back to  HYPERLINK \l "_Question_8." question 8. Link back to  HYPERLINK \l "_Question_9." question 9. Explanation for  HYPERLINK \l "_Question_9." question 9. (Follow link back to original question.) Program: SAT, PSAT/NMSQT, PSAT 10 Subscore: Heart of algebra Calculator usage: Yes Objective: Students must convert an existing compound inequality into a different format and find a value that satisfies all conditions. Difficulty: Easy Key: Any value greater than  EMBED Equation.DSMT4  the fraction 21 over 4 and less than  EMBED Equation.DSMT4  the fraction 27over5 Recognizing the structure of this inequality provides one solution strategy. With this strategy, a student will look at the relationship between  EMBED Equation.DSMT4  negative3t plus1 and  EMBED Equation.DSMT4  9t minus3 and recognize that the latter is  EMBED Equation.DSMT4  negative3 multiplied by the former. Multiplying all parts of the inequality by  EMBED Equation.DSMT4  negative3 reverses the inequality signs, resulting in  EMBED Equation.DSMT4  the fraction 27over5 is greater than 9t minus3, and 9t minus3 is greater than the fraction 21over4, or rather  EMBED Equation.DSMT4  the fraction 21over4 is less than 9t minus3, and 9t minus3 is less than the fraction 27over5 when written with increasing values from left to right. Any value greater than  EMBED Equation.DSMT4  the fraction 21over4 and less than  EMBED Equation.DSMT4  the fraction 27over5 is correct. Link back to  HYPERLINK \l "_Question_9." question 9. Link back to  HYPERLINK \l "_Question_10." question 10. Explanation for  HYPERLINK \l "_Question_10." question 10. (Follow link back to original question.) Program: SAT, PSAT/NMSQT, PSAT 10 Subscore: Problem solving and data analysis Calculator usage: Yes Objective: Students must conceptualize the context and retrieve relevant information from a table, and then manipulate it to form or compare relevant quantities. Difficulty: Easy Key: C Choice C is the correct answer. The first question asks students to select the relevant information from  HYPERLINK \l "Title_2" the table (followlink) to compute the percentage of selfreported voters for each age group and then compare the percentages to identify the largest one, choice C. Of the 55- to 74yearold groups total population (59,998,000), 43,075,000 reported that they had voted, which represents 71.8% and is the highest percentage of reported voters from among the four agegroups. Choice A is not the correct answer. The question is asking for the age group with the largest percentage of selfreported voters. This answer reflects the age group with the smallest percentage of selfreported voters. This groups percentage of selfreported voters is 48.1%, or  EMBED Equation.DSMT4  the fraction 30,329 over63,008, which is less than that of the 55- to 74yearoldgroup. Choice B is not the correct answer. The question is asking for the age group with the largest percentage of selfreported voters. This answer reflects the age group with the largest number of selfreported voters, not the largest percentage. This groups percentage of selfreported voters is 63.4%, or  EMBED Equation.DSMT4  the fraction 47,085 over74,282, which is less than that of the 55- to 74yearoldgroup. Choice D is not the correct answer. The question is asking for the age group with the largest percentage of selfreported voters. This answer reflects the age group with the smallest number of selfreported voters, not the largest percentage. This groups percentage of selfreported voters is 70.0%, or  EMBED Equation.DSMT4 the fraction 12,459 over17,794, which is less than that of the 55- to 74yearoldgroup. Link back to  HYPERLINK \l "_Question_10." question 10. Link back to  HYPERLINK \l "_Question_11." question 11. Explanation for  HYPERLINK \l "_Question_11." question 11. (Follow link back to original question.) Program: SAT, PSAT/NMSQT, PSAT 10 Subscore: Problem solving and data analysis Calculator usage: Yes Objective: Students must extrapolate from a random sample to estimate a population parameter. Difficulty: Medium Key: D Choice D is the correct answer.This question asks students to extrapolate from a random sample to estimate the number of 18 to 34yearolds who voted for CandidateA: this is done by multiplying the fraction of people in the random sample who voted for CandidateA by the total population of voting 18 to 34yearolds:  EMBED Equation.DSMT4  the fraction 287 over500, multiplied by 30,329,000, is almost equal to 17million, choice D. Students without a clear grasp of the context and its representation in the table might easily arrive at one of the other answers listed. Choice A is not the correct answer. The student may not have multiplied the fraction of the sample by the correct subgroup of people (18 to 34yearolds who voted). This answer may result from multiplying the fraction by the entire population, which is an incorrect application of the information. Choice B is not the correct answer. The student may not have multiplied the fraction of the sample by the correct subgroup of people (18 to 34yearolds who voted). This answer may result from multiplying the fraction by the total number of people who voted, which is an incorrect application of the information. Choice C is not the correct answer. The student may not have multiplied the fraction of the sample by the correct subgroup of people (18 to 34yearolds who voted). This answer may result from multiplying the fraction by the total number of 18 to 34yearolds, which is an incorrect application of the information. Link back to  HYPERLINK \l "_Question_11." question 11. Link back to  HYPERLINK \l "_Question_12." question 12. Explanation for  HYPERLINK \l "_Question_12." question 12. (Follow link back to original question.) Program: SAT, PSAT/NMSQT, PSAT 10 Subscore: Heart of algebra Calculator usage: Yes Objective: Students must interpret an expression or equation that models a realworld situation and be able to interpret the whole expression (or specific parts) in terms of its context. Difficulty: Medium Key: C Choice C is correct. One method to find the correct answer is to create an inequality. The income from sales of n items is 12n. For the company to profit, 12n must be greater than the cost of producing n items; therefore, the inequality  EMBED Equation.DSMT4  12n is greater than 7n plus350 can be used to model the context. Solving this inequality yields  EMBED Equation.DSMT4  n is greater than70. Choice A is not the correct answer. This answer may result from a misunderstanding of the properties of inequalities. The student may have found the number of items of the breakeven point as 70and used the incorrect notation to express the answer, or the student may have incorrectly modeled the scenario when setting up an inequality to solve. Choice B is not the correct answer. This answer may result from a misunderstanding of how the cost equation models the scenario. A student who uses the cost of $12 as the number of items n and evaluates the expression7n will find the value of 84. A student who does not understand how the inequality relates to the scenario may think n should be less than this value. Choice D is not the correct answer. This answer may result from a misunderstanding of how the cost equation models the scenario. A student who uses the cost of $12 as the number of items n and evaluates the expression7n will find the value of 84. A student who does not understand how the inequality relates to the scenario may think n should be greater than this value. Link back to  HYPERLINK \l "_Question_12." question 12. Link back to  HYPERLINK \l "_Question_13." question 13. Explanation for  HYPERLINK \l "_Question_13." question 13. (Follow link back to original question.) Program: SAT, PSAT/NMSQT, PSAT 10 Subscore: Problem solving and data analysis Calculator usage: Yes Objective: Students must evaluate the means for two separate populations in order to determine the constraints on the mean for the combined population. Difficulty: Medium Key: D Choice D is correct. The student must reason that because the mean of the males is lower than that of the females, the combined mean cannot be greater than or equal to that of the females, while also reasoning that because the mean of the females is greater than that of the males, the combined mean cannot be less than or equal to the mean of the males. Therefore the combined mean must be between the two separate means. Choice A is not the correct answer. This answer results from a student finding the mean of the two means. This answer makes an unjustified assumption that there are an equal number of male and female primates. Choice B is not the correct answer. This answer results from a student finding the mean of the two means and misapplying an inequality to the scenario. This answer makes an unjustified assumption that there are more females than males. Choice C is not the correct answer. This answer results from a student finding the mean of the two means and misapplying an inequality to the scenario. This answer makes an unjustified assumption that there are more males than females. Link back to  HYPERLINK \l "_Question_13." question 13. Link back to  HYPERLINK \l "_Question_14." question 14. Explanation for  HYPERLINK \l "_Question_14." question 14. (Follow link back to original question.) Program: SAT Subscore: Problem solving and data analysis Calculator usage: Yes Objective: Students must use information from a research study to evaluate whether the results can be generalized to the study population and whether a causeandeffect relationship exists. To conclude a causeandeffect relationship like the ones described in choiceC and choiceD, there must be random assignment of participants to groups receiving different treatments. To conclude that the relationship applies to a population, participants must be randomly selected from that population. Difficulty: Medium Key: A Choice A is correct. A relationship in the data can only be generalized to the population that the sample was drawn from. Choice B is not the correct answer. A relationship in the data can only be generalized to the population that the sample was drawn from. The sample was from high school students in the UnitedStates, not from high school students in the entireworld. Choice C is not the correct answer. Evidence for a causeandeffect relationship can only be established when participants are randomly assigned to groups that receive different treatments. Choice D is not the correct answer. Evidence for a causeandeffect relationship can only be established when participants are randomly assigned to groups that receive different treatments. Also, a relationship in the data can only be generalized to the population that the sample was drawn from. The sample was from high school students in the UnitedStates, not from high school students in the entire world. Link back to  HYPERLINK \l "_Question_14." question 14. Link back to  HYPERLINK \l "_Question_15." question 15. Explanation for  HYPERLINK \l "_Question_15." question 15. (Follow link back to original question.) Program: SAT, PSAT/NMSQT, PSAT 10 Subscore: Passport to advanced math Calculator usage: Yes Objective: Students must identify the correct mathematical notation for an exponential relationship that represents a realworld situation. Difficulty: Medium Key: D Choice D is correct. The student first recognizes that a population that doubles in size over equal time periods is increasing at an exponential rate. An exponential growth model can be written in the form  EMBED Equation.DSMT4  y equals a, parenthesis 2, close parenthesis, raised to the power the fraction x overb, where a is the population at time 0, since 2 raised to the zeroth power is 1 and b is the doubling time, since when  EMBED Equation.DSMT4  xequals12, 2 is raised to the first power, and the population will be  EMBED Equation.DSMT4  2 raised to the power ofone, times a, equals2a. From the way the variables were defined, the population at time  EMBED Equation.DSMT4  n equals0 is 50 and the doubling time is12. Choice A is not the correct answer. This answer may result from a misunderstanding of exponential equations or of the context. A student who tries to model the scenario with a linear equation or who misunderstands that the yintercept of a linear model should represent the initial number of animals may produce this equation. Choice B is not the correct answer. This answer may result from a misunderstanding of exponential equations or of the scenario. A student who tries to model the scenario with a linear equation may produce this equation. Choice C is not the correct answer. A student who tries to produce an exponential model, but does not understand how the 12years affects the model, may incorrectly write the exponent. Link back to  HYPERLINK \l "_Question_15." question 15. Link back to  HYPERLINK \l "_Question_16._1" question 16. Explanation for  HYPERLINK \l "_Question_16._1" question 16. (Follow link back to original question.) Program: SAT, PSAT/NMSQT, PSAT 10 Subscore: No subscore (additional topics in math) Calculator usage: Yes Objective: Students must use spatial reasoning and geometric logic to deduce which relationship is possible based on the given information. Students must also use mathematical notation to express the relationship between the line segments. Difficulty: Medium Key: C Choice C is correct. Given that  EMBED Equation.DSMT4  triangle ABC is similar to  EMBED Equation.DSMT4  triangleEDC, students can determine that the corresponding angle BAC is congruent to angle CED. The converse of the alternate interior angle theorem tells us that EMBED Equation.DSMT4  line segment AB is parallel to line segmentDE. (Students can also use the fact that  EMBED Equation.DSMT4  angle ABC and  EMBED Equation.DSMT4  angle CDE are congruent to make a similar argument.) Choice A is not the correct answer. This answer may result from multiple misconceptions. The student may have visually identified the segments as perpendicular and used the wrong notation to express this statement. Choice B is not the correct answer. This answer may result from visual inspection of the diagram. The line segments appear to be perpendicular, but need not be, given the information provided. Choice D is not the correct answer. This answer may result from misunderstanding either the notation or the vocabulary of parallel and perpendicular lines. The student has incorrectly identified or notated parallel lines as perpendicular. Link back to  HYPERLINK \l "_Question_16._1" question 16. Link back to  HYPERLINK \l "_Question_17." question 17. Explanation for  HYPERLINK \l "_Question_17." question 17. (Follow link back to original question.) Program: SAT, PSAT/NMSQT, PSAT 10 Subscore: Heart of algebra Calculator usage: Yes Objective: Students must create a function to represent a realworld situation. Difficulty: Medium Key: B Choice B is correct. Since Peters car is traveling at an average speed of 50 miles per hour and the cars gas mileage is 21 miles per gallon, the number of gallons of gas used each hour can be found by  EMBED Equation.DSMT4  the fraction 50miles over 1hour, times the fraction 1gallon over 21miles, equals the fraction 50 over21. The car uses  EMBED Equation.DSMT4  the fraction 50 over 21 gallons of gas per hour, so it uses  EMBED Equation.DSMT4  the fraction 50 over 21, timest,gallons of gas in thours. The cars gas tank has 17gallons of gas at the beginning of the trip. Therefore, the function that models the number of gallons of gas remaining in the tank thours after the trip begins is  EMBED Equation.DSMT4  f oft equals 17 minus the fraction 50t over21. Choice A is not the correct answer. The number of gallons of gas used each hour is determined by dividing the average speed by the cars gas mileage. Choice C is not the correct answer. The number of gallons of gas used each hour is misrepresented as  EMBED Equation.DSMT4  the fraction 21 over50. Also, the number of gallons used each hour must be multiplied by timet before it is subtracted from the number of gallons of gas in the tank at the beginning of the trip. Choice D is not the correct answer. The number of gallons of gas used each hour must be multiplied by timet before it is subtracted from the number of gallons of gas in the tank at the beginning of the trip Link back to  HYPERLINK \l "_Question_17." question 17. Link back to  HYPERLINK \l "_Question_18." question 18. Explanation for  HYPERLINK \l "_Question_18." question 18. (Follow link back to original question.) Program: SAT, PSAT/NMSQT, PSAT 10 Subscore: Heart of algebra Calculator usage: Yes Objective: Students must create a system of linear equations to represent a realworld situation. Difficulty: Medium Key: C Choice C is correct. If x is the number of cars that crossed the bridge during the two hours andy is the number of trucks that crossed the bridge during the two hours, then  EMBED Equation.DSMT4  xplusy represents the total number of cars and trucks that crossed the bridge during the two hours, and  EMBED Equation.DSMT4  6.5x plus 10y represents the total amount collected in the two hours. Therefore, the correct system of equations is  EMBED Equation.DSMT4  xplusy equals187 and  EMBED Equation.DSMT4  6.5x plus 10y equals1,338. Choice A is not the correct answer. The student may have mismatched the symbolic expressions for total cars and trucks and total tolls collected with the two numerical values given. The expression  EMBED Equation.DSMT4  xplusy represents the total number of cars and trucks that crossed the bridge, which is 187. Choice B is not the correct answer. The student may have attempted to use the information that the counts of cars, trucks, and tolls were taken over a period of two hours, but this information is not needed in setting up the correct system of equations. The expression  EMBED Equation.DSMT4  6.5x plus 10y represents the total amount of tolls collected, which is $1,338, not  EMBED Equation.DSMT4  the fraction 1,338dollars over2. Choice D is not the correct answer. The student may have attempted to use the information that the counts of cars, trucks, and tolls were taken over a period of two hours, but this information is not needed in setting up the correct system of equations. The expression  EMBED Equation.DSMT4  6.5x plus 10y represents the total amount of tolls collected, which is $1,338, not  EMBED Equation.DSMT4  1,338dollars times2. Link back to  HYPERLINK \l "_Question_18." question 18. Link back to  HYPERLINK \l "_Question_19." question 19. Explanation for  HYPERLINK \l "_Question_19." question 19. (Follow link back to original question.) Program: SAT, PSAT/NMSQT, PSAT 10 Subscore: Heart of algebra Calculator usage: Yes Objective: Students must construct a linear equation to represent a realworld situation. Difficulty: Medium Key: B Choice B is correct. To determine the linear model, one can first determine the rate at which the pressure due to the atmosphere and surrounding water is increasing as the depth of the diver increases. Calculating this gives  EMBED Equation.DSMT4  the fraction whose numerator is 20.9 minus 18.7, and whose denominator is 14minus9, equals, the fraction 2.2 over5, or 0.44. Then one needs to determine the pressure due to the atmosphere or, in other words, the pressure when the diver is at a depth of 0. Solving the equation  EMBED Equation.DSMT4  18.7 equals, 0.44, parenthesis, 9, close parenthesis, plusb gives  EMBED Equation.DSMT4  bequals 14.74. Therefore, the model that can be used to relate the pressure and the depth is  EMBED Equation.DSMT4  pequals 0.44d plus 14.74. Choice A is not the correct answer. The rate is calculated correctly, but the student may have incorrectly used the ordered pair  EMBED Equation.DSMT4  parenthesis, 18.7 comma 9, close parenthesis, rather than  EMBED Equation.DSMT4  parenthesis, 9comma 18.7, close parenthesis, to calculate the pressure at a depth of 0feet. Choice C is not the correct answer. The rate here is incorrectly calculated by subtracting 20.9 and 18.7 and not dividing by5. The student then uses the coordinate pair  EMBED Equation.DSMT4  dequals9 and  EMBED Equation.DSMT4  p equals 18.7 inconjunction with the incorrect slope of 2.2 to write the equation of the linearmodel. Choice D is not the correct answer. The rate here is incorrectly calculated by subtracting 20.9 and 18.7 and not dividing by5. The student then uses the coordinate pair  EMBED Equation.DSMT4  dequals14 and  EMBED Equation.DSMT4  p equals 20.9 inconjunction with the incorrect slope of 2.2 to write the equation of the linearmodel. Link back to  HYPERLINK \l "_Question_19." question 19. Link back to  HYPERLINK \l "_Question_20." question 20. Explanation for  HYPERLINK \l "_Question_20." question 20. (Follow link back to original question.) Program: SAT, PSAT/NMSQT, PSAT 10 Subscore: Problem solving and data analysis Calculator usage: Yes Objective: Students must interpret the slope of the line of best fit for the scatterplot as the average increase in the number of manatees per year, while taking the scales of the axes into consideration. Difficulty: Medium Key: C Choice C is correct. The slope of the line of best fit is the value of the average increase in manatees per year. Using approximate values found along the line of best fit (1,200manatees in1991 and 4,200manatees in2011), the approximate slope can be calculated as  EMBED Equation.DSMT4  the fraction 3,000 over 20 equals150. Choice A is not the correct answer. This value may result from disregarding the actual scale when approximating the slope and interpreting the scale as if each square represents oneunit. Choice B is not the correct answer. This value may result from disregarding the actual scale when approximating the slope, and interpreting the scale as if each square along the xaxis represents one year and each tick mark along the yaxis represents 100manatees. Choice D is not the correct answer. This value may result from disregarding the actual scale along the xaxis when approximating the slope and interpreting each square along the xaxis as oneyear. Link back to  HYPERLINK \l "_Question_20." question 20. Link back to  HYPERLINK \l "_Question_21." question 21. Explanation for  HYPERLINK \l "_Question_21." question 21. (Follow link back to original question.) Program: SAT, PSAT/NMSQT, PSAT 10 Subscore: Problem solving and data analysis Calculator usage: Yes Objective: Students must synthesize graphical and textual information and determine what information is relevant to the solution. Difficulty: Medium Key: B Choice B is the correct answer. Each petri dish has area 10squarecentimeters, and so at time  EMBED Equation.DSMT4 tequalszero, Dish1 is 10% covered  EMBED Equation.DSMT4  parenthesis, onetenth, close parenthesis, and Dish2 is 20% covered  EMBED Equation.DSMT4  parenthesis, twotenths, close parenthesis. Thus the statement in choiceB istrue. Choice A is not the correct answer. At the end of the observations, both dishes are 100% covered with bacteria, but at time  EMBED Equation.DSMT4 tequalszero, neither dish is 100%covered. Choice C is not the correct answer. At time  EMBED Equation.DSMT4 tequalszero, Dish1 is covered with 50% less bacteria than is Dish2, but Dish2 is covered with 100% more, not 50% more, bacteria than is Dish1. Choice D is not the correct answer. After the first hour, it is still true that more of Dish2 is covered by bacteria than is Dish1, but for the first hour the area of Dish1 that is covered has been increasing at a higher average rate (about 0.8squarecentimeterperhour) than the area of Dish2 (about 0.1squarecentimeterperhour). Link back to  HYPERLINK \l "_Question_21." question 21. Link back to  HYPERLINK \l "_Question_22." question 22. Explanation for  HYPERLINK \l "_Question_22." question 22. (Follow link back to original question.) Program: SAT, PSAT/NMSQT, PSAT 10 Subscore: Problem solving and data analysis Calculator usage: Yes Objective: Students must use the unit rate (datatransmission rate) and the conversion between gigabits and megabits as well as conversions in units oftime. Difficulty: Medium Key: B Choice B is correct. The tracking station can receive 118,800megabits each day  EMBED Equation.DSMT4  parenthesis, the fraction 3megabits over 1second, times, the fraction 60seconds over 1minute, times, the fraction 60minutes over 1hour, times, 11hours, close parenthesis, which is about 116gigabits each day EMBED Equation.DSMT4  parenthesis, the fraction 118,800 over1,024, close parenthesis. If each image is 11.2 gigabits, then the number of images that can be received each day is  EMBED Equation.DSMT4  the fraction 116 over 11.2 is almost equal to 10.4. Since the question asks for the maximum number of typical images, rounding the answer down to 10 is appropriate because the tracking station will not receive a completed eleventh image in one day. Choice A is not the correct answer. The student may not have synthesized all of the information. This answer may result from multiplying 3 (rate in megabitspersecond) by 11 (hours receiving) and dividing by 11.2 (size of image in gigabits), neglecting to convert 3megabits persecond into megabits per hour and to utilize the information about 1gigabit equaling 1,024megabits. Choice C is not the correct answer. The student may not have synthesized all of the information. This answer may result from converting the number of gigabits in an image to megabits (11,470), multiplying by the rate of 3megabits persecond (34,410), and then converting 11hours into minutes (660) instead ofseconds. Choice D is not the correct answer. The student may not have synthesized all of the information. This answer may result from converting 11hours into seconds (39,600), then dividing the result by 3gigabits converted into megabits (3,072), and multiplying by the size of one typicalimage. Link back to  HYPERLINK \l "_Question_22." question 22. Link back to  HYPERLINK \l "_Question_23." question 23. Explanation for  HYPERLINK \l "_Question_23." question 23. (Follow link back to original question.) Program: SAT Subscore: Passport to advanced math Calculator usage: Yes Objective: Students must manipulate one equation for use in another, in order to solve for a given value, making use of familiar algebraic arrangements where appropriate. Difficulty: Medium Key: C Choice C is correct. The second equation gives y in terms of x, so a student can use this to rewrite the first equation in terms of x. Substituting  EMBED Equation.DSMT4  negative4x for y in the equation  EMBED Equation.DSMT4  xsquared plus ysquared equals153 gives  EMBED Equation.DSMT4  xsquared plus, parenthesis, negative4x, close parenthesis, squared, equals153. This can be simplified to  EMBED Equation.DSMT4  xsquared plus 16xsquared equals153, or  EMBED Equation.DSMT4  17xsquared equals153. Since the question asks for the value of  EMBED Equation.DSMT4  xsquared, not x, dividing both sides of  EMBED Equation.DSMT4  17xsquared equals153 by 17 gives the answer:  EMBED Equation.DSMT4  xsquared equals the fraction 153over17, equals9. Choice A is not the correct answer. This answer may result from neglecting to square the coefficient  EMBED Equation.DSMT4  negative4 in  EMBED Equation.DSMT4  yequals negative4x, which would give  EMBED Equation.DSMT4  y squared equals negative4xsquared. Then the first equation would become  EMBED Equation.DSMT4  xsquared minus 4xsquared, equals negative3xsquared, equals153, which would give  EMBED Equation.DSMT4  negative51 as the value of  EMBED Equation.DSMT4  xsquared. Choice B is not the correct answer. This answer may result from finding the value forx, not the value of  EMBED Equation.DSMT4  xsquared. Choice D is not the correct answer. This answer may result from finding the value of  EMBED Equation.DSMT4  ysquared, not  EMBED Equation.DSMT4  xsquared. Link back to  HYPERLINK \l "_Question_23." question 23. Link back to  HYPERLINK \l "_Question_24." question 24. Explanation for  HYPERLINK \l "_Question_24." question 24. (Follow link back to original question.) Program: SAT Subscore: No subscore (additional topics in math) Calculator usage: Yes Objective: Students must make connections between physical concepts such as mass and density and essential geometric ideas such as the Pythagorean theorem and volume formulas. Difficulty: Medium Key: 57 This question asks students to make connections between physical concepts such as mass and density and essential geometric ideas such as the Pythagoreantheorem and volume formulas. There are multiple approaches to solving this problem, but in any of them, the aim is to find the volume of the metal nut and then use the density of the metal to calculate the mass of the nut (57grams). This is a multistep problem that requires students to devise a multistep strategy and carry out all the algebraic and numerical steps without error. To solve this problem, students need to find the volume of the hex nut and then use the given fact that density is mass divided by volume. Finding the volume of the hex nut requires several steps. The first step is to calculate the area of one of the hexagonal faces (without the drilled hole). Each face is a regular hexagon, which can be divided into 6 equilateral triangles with side lengths of 2centimeters. Using 306090 triangle properties, the height of each equilateral triangle is  EMBED Equation.DSMT4  the squareroot ofthree centimeters. In turn, the area of one equilateral triangle is  EMBED Equation.DSMT4  onehalfbh equals onehalf, parenthesis, 2, close parenthesis, parenthesis, the squareroot of 3, close parenthesis, equals the square root of 3 square centimeters, so the area of the hexagonal face is  EMBED Equation.DSMT4  6 times the square root of 3, square centimeters. The volume of the hexagonal prism is the area of one face multiplied by the height (or thickness in this case),  EMBED Equation.DSMT4  parenthesis, 6 times the square root of 3, close parenthesis, parenthesis, 1, close parenthesis, equals 6 times the square root of3, cubiccentimeters. Then, to account for the drilled hole, students need to calculate the volume of a cylinder with diameter 2centimeters (or radius 1centimeter) and height onecentimeter,  EMBED Equation.DSMT4  capital V equals pi rsquared, h, equals, pi, parenthesis, 1 squared, close parenthesis, parenthesis, 1, close parenthesis, equalspi, cubic centimeter, and subtract it from the volume of the hexagonal prism to yield  EMBED Equation.DSMT4  6 times the square root of3, minus pi, cubic centimeter. Finally, density is mass divided by volume,  EMBED Equation.DSMT4  7.9, equals the fraction whose numerator is mass and whose denominator is 6 times the square root of 3, minus pi. Multiplying both sides of the equation by  EMBED Equation.DSMT4  parenthesis, 6 times the square root of 3, minus pi, close parenthesis, yields the mass of the hex nut as  EMBED Equation.DSMT4  7.9, parenthesis, 6 times the square root of 3, minus pi, close parenthesis, grams. When the values for  EMBED Equation.DSMT4  the square root of 3 and  EMBED Equation.DSMT4  pi are substituted and the result is rounded to the nearest gram, the answer is approximately 57grams. Note that it is critical for students to attend to the precision of their calculations when solving this problem, and not apply any intermediate rounding until the final answer is reached. Here, the use of a calculator provides the ability to attend to precision more effectively, and thus is highly encouraged. Link back to  HYPERLINK \l "_Question_24._1" question 24. Link back to  HYPERLINK \l "_Question_25." question 25. Explanation for  HYPERLINK \l "_Question_25." question 25. (Follow link back to original question.) Program: SAT, PSAT/NMSQT, PSAT 10 Subscore: Problem solving and data analysis Calculator usage: Yes Objective: Students must use a linear equation to determine the conversion rate between two currencies. Difficulty: Medium Key: 63 $9.88 represents the conversion of 602rupees plus a 4%fee on the converted cost.To calculate the original cost of the item in dollars, x:  EMBED Equation.DSMT4  1.04x equals, 9.88.  EMBED Equation.DSMT4  x equals 9.5. Since the original cost is $9.50, to calculate the exchange rater, in Indianrupees per oneU.S.dollar:  EMBED Equation.DSMT4  9dollars and 50cents, times the fraction rrupees over onedollar, equals 602rupees.  EMBED Equation.DSMT4  r equals, the fraction 602 over 9.50.  EMBED Equation.DSMT4  is almost equal to 63rupees. Link back to  HYPERLINK \l "_Question_25." question 25. Link back to  HYPERLINK \l "_Question_26." question 26. Explanation for  HYPERLINK \l "_Question_26." question 26. (Follow link back to original question.) Program: SAT, PSAT/NMSQT, PSAT 10 Subscore: Problem solving and data analysis Calculator usage: Yes Objective: Students must use the currency conversion information and the constraints provided to determine a meaningful value in the context of the problem. Difficulty: Hard Key: 7,212 Let d dollars be the cost of the 7,500rupee prepaid card. This implies that the exchange rate on this particular day is  EMBED Equation.DSMT4  the fraction d over 7,500 dollars per rupee. Suppose Saras total purchases on the prepaid card were rrupees. The value of the rrupees in dollars is  EMBED Equation.DSMT4  parenthesis, the fraction d over 7,500, close parenthesis, r, dollars. If Sara spent the rrupees on the Traveler card instead, she would be charged  EMBED Equation.DSMT4  parenthesis, 1.04, close parenthesis, parenthesis, the fraction d over 7,500, close parenthesis, r, dollars. To answer the question about how many rupees Sara must spend in order to make the Traveler card a cheaper option (in dollars) for spending the rrupees, we set up the inequality  EMBED Equation.DSMT4  1.04, parenthesis, the fraction d over 7,500, close parenthesis, r is greater than or equal to d. Rewriting both sides reveals  EMBED Equation.DSMT4  1.04, parenthesis, the fraction r over 7,500, close parenthesis, d is greater than or equal to, parenthesis, 1, close parenthesis, d, from which we can infer  EMBED Equation.DSMT4  1.04, parenthesis, the fraction r over 7,500, close parenthesis, is greater than or equal to1. Dividing on both sides by 1.04 and multiplying on both sides by 7,500 finally yields  EMBED Equation.DSMT4  r is greater than or equal to7,212. Hence the least number of rupees Sara must spend for the prepaid card to be cheaper than the Traveler card is7,212. Link back to  HYPERLINK \l "_Question_26." question 26. Link back to  HYPERLINK \l "_Question_27." question 27. Explanation for  HYPERLINK \l "_Question_27." question 27. (Follow link back to original question.) Program: SAT, PSAT/NMSQT, PSAT 10 Subscore: Heart of algebra Calculator usage: Yes Objective: Students must understand the relationship between an equation in two variables and the characteristics of its graph (for example, shape, position, intercepts, extreme points, or symmetry). In addition, students must transform the given equation into a more suitable form and then make the connection between the obtained equation and the graph. Difficulty: Hard Key: B Choice B is correct. Manipulating the equation to solve fory gives  EMBED Equation.DSMT4  yequals, parenthesis, the fraction whose numerator is 1 plus k and whose denominator is 1 minus k, close parenthesis, x, revealing that the graph of the equation must be a line that passes through the origin. Of the choices given, only the graph shown in choice B satisfies these conditions. Choice A is not the correct answer. The student may have seen that the term  EMBED Equation.DSMT4  k, parenthesis, x plus y, close parenthesis, is a multiple of  EMBED Equation.DSMT4  xplusy and wrongly concluded that this is the equation of a line with slope1. Choice C is not the correct answer. The student may have made incorrect steps when simplifying the equation or may have not seen the advantage that putting the equation in slopeintercept form would give in determining the graph, and thus wrongly concluded the graph has a nonzero yintercept. Choice D is not the correct answer. The student may not have seen that term  EMBED Equation.DSMT4  k, parenthesis, x plus y, close parenthesis, can be multiplied out and the variables x and y isolated, and wrongly concluded that the graph of the equation cannot be a line. Link back to  HYPERLINK \l "_Question_27." question 27. Link back to  HYPERLINK \l "_Question_28." question 28. Explanation for  HYPERLINK \l "_Question_28." question 28. (Follow link back to original question.) Program: SAT Subscore: Passport to advanced math Calculator usage: Yes Objective: Students must understand the zeros of a polynomial function and how they are used to construct algebraic representations of polynomials. Difficulty: Hard Key: A Choice A is correct. The given zeros can be used to set up an equation to solve forc. Substituting  EMBED Equation.DSMT4  negative4 for x and 0 for y yields  EMBED Equation.DSMT4  negative4c equals72, or  EMBED Equation.DSMT4  c equals negative18. Alternatively, since  EMBED Equation.DSMT4  negative4,  EMBED Equation.DSMT4  onehalf, and p are zeros of the polynomial function  EMBED Equation.DSMT4  fofx equals 2xcubed, plus 3xsquared, plus cx, plus8, it follows that  EMBED Equation.DSMT4  fofx equals, parenthesis, 2xminus 1, close parenthesis, parenthesis, x plus 4, close parenthesis, parenthesis x minus p, close parenthesis. Were this polynomial multiplied out, the constant term would be  EMBED Equation.DSMT4  parenthesis, negative1, close parenthesis, parenthesis, 4, close parenthesis, parenthesis negativep, close parenthesis, equals 4p. (We can see this without performing the full expansion.) Since it is given that this value is 8, it follows that  EMBED Equation.DSMT4  4p equals8 or rather,  EMBED Equation.DSMT4  p equals2. Substituting 2 for p in the polynomial function yields  EMBED Equation.DSMT4  fofx equals, parenthesis, 2x minus 1, close parenthesis, parenthesis, x plus 4, close parenthesis, parenthesis, xminus2, close parenthesis, and after multiplying the factors one finds that the coefficient of the x term, or the value of c, is  EMBED Equation.DSMT4  negative18. Choice B is not the correct answer. This value is a misunderstood version of the value of p, not c, and the relationship between the zero and the factor (if a is the zero of a polynomial, its corresponding factor is  EMBED Equation.DSMT4  xminusa) has been confused. Choice C is not the correct answer. This is the value of p, not c. Using this value as the third factor of the polynomial will reveal that the value of c is  EMBED Equation.DSMT4  negative18. Choice D is not the correct answer. This represents a sign error in the final step in determining the value ofc. Link back to  HYPERLINK \l "_Question_28." question 28. Link back to  HYPERLINK \l "_Question_29." question 29. Explanation for  HYPERLINK \l "_Question_29." question 29. (Follow link back to original question.) Program: SAT Subscore: Passport to advanced math Calculator usage: Yes Objective: Students must transform a given expression into a more useful form (from improper to proper rational form). Difficulty: Hard Key: A Choice A is correct. The form of the equation suggests performing long division on  EMBED Equation.DSMT4  the fraction whose numerator is 4xsquared, and whose denominator is 2xminus1:  EMBED Equation.DSMT4  Divide 2xminus1 into 4x squared, place 2xplus 1 on top of the long division symbol. Multiply 2x plus 1 by 2x, yields 4x squared minus 2x. Then divide 2xminus 1 into 2x, place positive 1 on top of the long division symbol preceded by 2x. Multiply 2xminus 1 by positive 1, yields 2xminus1. Subtract 2xminus 1 from 2x, yields the remainder1. Since the remainder 1 matches the numerator in  EMBED Equation.DSMT4  the fraction whose numerator is 1 and whose denominator is 2 x minus1, it is clear that  EMBED Equation.DSMT4  Aequals 2x plus1. A short way to find the answer is to use the structure to rewrite thenumerator of the expression as  EMBED Equation.DSMT4  parenthesis, 4x squared minus 1, close parenthesis, plus1, recognizing the term inparentheses as a difference of squares, making the expression equal to  EMBED Equation.DSMT4  the fraction whose numerator is, parenthesis, 2x minus 1, close parenthesis, parenthesis, 2x plus 1, close parenthesis, plus 1, and whose denominator is 2x minus 1, equals, 2x plus 1, plus the fraction whose numerator is 1 and whose denominator is 2x minus1. From this, the answer  EMBED Equation.DSMT4  2x plus1 is apparent. Another way to find the answer is to isolateAin the form  EMBED Equation.DSMT4  A equals the fraction whose numerator is 4x squared, and whose denominator is 2x minus 1, minus, the fraction whose numerator is 1, and whose denominator is 2x minus 1, and simplify. As with the first approach, this approach alsorequires students to recognize  EMBED Equation.DSMT4  4xsquared minus1 as a difference of squares that factors. Choice B is not the correct answer. The student may have made a signerror while subtracting partial quotients in the long division. Choice C is not the correct answer. The student may misunderstandhow to work with fractions and may have tried the incorrect calculation  EMBED Equation.DSMT4 the fraction whose numerator is 4xsquared, and whose denominator is 2xminus 1, equals, the fraction whose numerator is, parenthesis, 1, close parenthesis, parenthesis, 4xsquared, close parenthesis, and whose denominator is 2xminus 1, equals, the fraction whose numerator is 1, and whose denominator is 2xminus 1, plus 4xsquared. Choice D is not the correct answer. The student may misunderstand how to work with fractions and may have tried the incorrect calculation  EMBED Equation.DSMT4  the fraction whose numerator is 4xsquared, and whose denominator is 2xminus 1, equals, the fraction whose numerator is 1 plus 4xsquared minus ,, and whose denominator is 2xminus 1, equals, the fraction whose numerator is 1, and whose denominator is 2xminus 1, plus, 4xsquared, minus1. Link back to -. ,4D#&0d*+UXh- h8hF h-hFh}hF0JOJQJ h}hFhYFh&_h&_EHhFPJnHo(tHhFPJnHtH hF5 hhFh/hFEHh-hFCJaJhF7.\ p H ? A G %^  ^ `gd\ x^ `gd\gdFgdYFdgdFgdFgdFgdFgdFgdF+"X_0W ^  ^ `gdcgd gddqgdgddqgddqgd-gd-gdYFgdFgdF3./0 !¾ԾԾԾԾԾԤԕԕh>hdq6@h>hdq6@2h>Qht^@(B*aJph# hhdqh h4yh huh-h hhFhYF hKhdqhdqhFPJnHo(tHhFPJnHtHh5whF5hF h%5khF2!!!!!!!!!""" 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HYPERLINK \l "_Question_29." question 29. Link back to  HYPERLINK \l "_Question_30." question 30. Explanation for  HYPERLINK \l "_Question_30." question 30. (Follow link back to original question.) Program: SAT Subscore: No subscore (additional topics in math) Calculator usage: Yes Objective: Students must make use of properties of triangles to solve a problem. Difficulty: Hard Key:  EMBED Equation.DSMT4  twothirds Because the triangle is isosceles, constructing a perpendicular from the top vertex to the opposite side will bisect the base and create two smaller right triangles. In a right triangle, the cosine of an acute angle is equal to the length of the side adjacent to the angle divided by the length of the hypotenuse. This gives  EMBED Equation.DSMT4  cosinex equals the fraction 16over24, which can be simplified to  EMBED Equation.DSMT4  cosinex equals the fraction 2over3. Link back to  HYPERLINK \l "_Question_30." question 30. 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