ÐÏà¡±á>þÿ <>þÿÿÿ;€ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿì¥ÁG ¿GbjbjŽÙŽÙ +dì³ì³y<Hÿÿÿÿÿÿ]&&&&&&&¦¦¦¦8Þ4,¦Ü¬J¢ìììììììòôôô/#9O$ˆ ô|`si&ìììììs`&&ììJ```ì"&ì&ìò:6p6&&&&ìò`’`ò&&òì> 5©@`½¦¦RòCHAPTER 29PARTICLES AND WAVES CONCEPTUAL QUESTIONS _____________________________________________________________________________________________ 1. REASONING AND SOLUTION A monochromatic light source emits photons of a single frequency. According to Equation 29.2, the energy, E, of a single photon is related to its frequency, f, by the relation EMBED "Equation" "Word Object1" \* mergeformat , where h is Planck's constant. The photons emitted by a source of light do not all have the same energy. Since the photons do not all have the same energy, then, from Equation 29.2, we can conclude that the photons do not all have the same frequency. Therefore, the source is not monochromatic. _____________________________________________________________________________________________ 2. REASONING AND SOLUTION According to the data given in Example 1, Chapter 24, the frequency of visible light ranges from EMBED "Equation" "Word Object2" \* mergeformat (red light) to EMBED "Equation" "Word Object2" \* mergeformat (violet light). According to Equation 29.2, the energy, E, of a photon is related to its frequency, f, by the relation EMBED "Equation" "Word Object1" \* mergeformat , where h is Planck's constant. According to Equation 29.2, the energy of a photon is directly proportional to its frequency. a. The red-colored light bulb emits photons with the lowest frequency compared to light bulbs of other colors (orange, yellow, green, or blue); therefore, the red-colored light bulb emits photons with the lowest energy. b. The color blue appears next to violet in the continuous visible spectrum; therefore, the frequency of blue light is slightly smaller than that of violet, but greater than the frequency of other colors of the visible spectrum. Thus, the blue-colored light bulb emits photons with the highest frequency compared to the other light bulbs; therefore, the blue-colored light bulb emits photons with the greatest energy. _____________________________________________________________________________________________ 3. REASONING AND SOLUTION A photon emitted by a higher-wattage red light bulb does not have more energy than a photon emitted by a lower-wattage red bulb. The wattage of a bulb describes the power output of a bulb. Since average power is defined as energy per unit time, the power output of a light bulb tells us the rate at which the light bulb produces energy. According to Equation 29.2, the energy, E, of a photon is related to its frequency, f, by the relation EMBED "Equation" "Word Object1" \* mergeformat , where h is Planck's constant. Thus, the energy of a photon depends only on the frequency of the associated light wave. The frequency of red light is the same, regardless of the rate of energy production; therefore, all photons of red light have the same energy, regardless of the nature of their source. Remark: Since the higher-wattage bulb provides more energy per unit time than the lower-wattage bulb, we can conclude that the higher-wattage bulb produces more photons per unit time. All of the "red photons," however, have the same energy. _____________________________________________________________________________________________ 4. REASONING AND SOLUTION When a sufficient number of visible light photons strike a piece of photographic film, the film becomes exposed. An X-ray photon is more energetic than a visible light photon. Yet, most photographic films are not exposed by the X-ray machines used at airport security checkpoints. Since a single X-ray photon is more energetic than a single photon of visible light, we can conclude that the number of X-ray photons per unit time emitted by airport security machines is much smaller than the number of visible light photons per unit time produced by normal lighting fixtures. _____________________________________________________________________________________________ 5. REASONING AND SOLUTION When radiation strikes a metallic surface, a photon of the radiation can give up its energy to an electron in the metal. If the photon has enough energy to perform the minimum amount of work required to remove the electron from the metal, the electron can be emitted from the metal. The minimum amount of work required to remove an electron from a metal is called its work function. If the photon does not have enough energy to remove an electron, the energy of the photon is absorbed by the electron; the increase in energy is manifested as thermal motion. Radiation of a given wavelength causes electrons to be emitted from the surface of one metal (metal 1) but not from the surface of another metal (metal 2). We can conclude that the individual photons of the radiation have sufficient energy to do the work required to remove the electrons from the surface of metal 1, but not enough energy to remove the electrons from the surface of metal 2. Therefore, the work function of metal 1 must be less than the work function of metal 2. _____________________________________________________________________________________________ 6. REASONING AND SOLUTION In a photoelectric effect experiment, the intensity of the light is increased while the frequency is kept constant. The frequency is greater than the minimum frequency EMBED "Equation" "Word Object23" \* mergeformat , so that photoelectrons are emitted from the negatively charged metal plate (see Figure 29.4). Light intensity is the energy per second per unit area that crosses a transverse surface. According to Einstein's photon hypothesis (see Equation 29.2), a beam of light is composed of individual photons, each having an energy EMBED "Equation" "Word Object24" \* mergeformat , where h is Planck's constant and f is the frequency of the light. From this viewpoint, the intensity of the light is equal to the energy of each photon multiplied by the number of photons per second per unit area crossing a transverse surface. An increase in the intensity of the light beam, therefore, corresponds to an increase in the number of photons per second that strikes the negatively charged metal plate. This will result in the emission of an increased number of photoelectrons per second at the negative plate. a. As the number of photoelectrons emitted per second from the negative plate increases, the number of photoelectrons collected per second at the positive plate increases. Since the current in the phototube is proportional to the number of photoelectrons per second collected at the positive plate, increasing the light intensity will cause the current in the phototube to increase. b. As discussed above, an increase in intensity will cause the number of photoelectrons emitted per second from the metal surface to increase. c. According to Equation 29.3, the maximum kinetic energy that a photoelectron could have is given by EMBED "Equation" "Word Object25" \* mergeformat , where EMBED "Equation" "Word Object26" \* mergeformat is the work function of the metal. Since the frequency is kept constant, and EMBED "Equation" "Word Object26" \* mergeformat is a property of the metal that is independent of the intensity, an increase in intensity will have no effect on EMBED "Equation" "Word Object27" \* mergeformat . The maximum kinetic energy that an electron could have remains the same. d. The kinetic energy of an object of mass m moving non-relativistically with speed v can be expressed in terms of its momentum p as follows: EMBED "Equation" "Word Object28" \* mergeformat Therefore, treating the electron classically, the maximum momentum that a photoelectron could have is EMBED "Equation" "Word Object29" \* mergeformat . Since the maximum kinetic energy that an electron could have remains the same as the intensity is increased at constant frequency, the maximum momentum that an electron could have remains the same as well. e. According to Equation 29.8, the minimum de Broglie wavelength of an object of maximum momentum EMBED "Equation" "Word Object31" \* mergeformat is given by EMBED "Equation" "Word Object30" \* mergeformat Since the maximum momentum that a photoelectron could have remains the same as the intensity is increased at constant frequency, we can conclude that the minimum de Broglie wavelength that an electron could have also remains the same. _____________________________________________________________________________________________ 7. REASONING AND SOLUTION As the result of a Compton scattering experiment, an electron is accelerated straight ahead in the same direction as that of the incident X-ray photon. Momentum conservation requires that the total initial momentum be equal to the total final momentum. The total initial momentum consists only of the forward momentum of the incoming photon, which we assume to be traveling in the +x direction. The electron is initially at rest. Therefore, the direction of the final total momentum of the recoiling electron and the scattered photon must also point in the +x direction. There can be no component of the final total momentum along the +y or –y direction. We know that the recoiling electron has momentum only in the +x direction. Thus, in order for the y component of the total final momentum to be absent, the scattered photon must move either in the +x or –x direction. To distinguish between these two choices, it is necessary to refer to energy conservation, as well as momentum conservation. _____________________________________________________________________________________________ 8. REASONING AND SOLUTION A photon can undergo Compton scattering from a molecule such as nitrogen, just as it does from an electron. However, the change in photon wavelength is much less than when an electron is scattered. To see why, let us examine Equation 29.7 which gives the difference between the wavelength lð' of the scattered photon and the wavelength lð of the incident photon in terms of the scattering angle qð: EMBED "Equation" "Word Object5" \* mergeformat , where h is Planck's constant, m is the mass of the target particle, and c is the speed of light in a vacuum. The mass of a nitrogen molecule is much greater than the mass of an electron. Therefore, the factor EMBED "Equation" "Word Object6" \* mergeformat will be much smaller if the target particle is a nitrogen molecule. Consequently, the change in the photon wavelength, lð' - lð, is much less than it is when the target particle is an electron. _____________________________________________________________________________________________ 9. REASONING AND SOLUTION When the speed of a particle with mass doubles, its momentum doubles, and its kinetic energy becomes four times greater. A photon, however, does not behave the same as a particle with mass. The momentum of a photon is EMBED "Equation" "Word Object32" \* mergeformat (see the derivation of Equation 29.6); therefore, the energy of the photon can be written as EMBED "Equation" "Word Object33" \* mergeformat . Clearly, when the momentum of a photon doubles, its energy also doubles. It does not become four times greater. _____________________________________________________________________________________________ 10. REASONING AND SOLUTION When bright light is incident on the radiometer, photons strike both the black and the shiny surfaces. The photons are absorbed by the black surfaces and are reflected by the shiny surfaces. Linear momentum is transferred to the panels because of the photon collisions. Since the orientation of the light is arbitrary, photons will strike the panels at arbitrary angles of incidence. Only the component of the photon's momentum that is perpendicular to the face of the panel contributes to its motion. As discussed in Conceptual Example 3, the momentum transfer to the panels is a maximum for the shiny surfaces and is twice as large in magnitude than it is for the black surfaces. Following the reasoning of Conceptual Example 3, we can use the impulse-momentum theorem and Newton's third law to deduce that the force on the shiny side is greater than the force on the black side. Therefore, in bright light, the arrangement would spin in the direction from the shiny side toward the black side. The observed spinning is in the opposite direction. Thus, photon collisions with the panels cannot be the cause of the spinning. _____________________________________________________________________________________________ 11. REASONING AND SOLUTION The linear momentum of an object of mass m traveling with velocity v is p = mv. The de Broglie wavelength of an object moving with momentum p is given by Equation 29.8: EMBED "Equation" "Word Object8" \* mergeformat , where h is Planck's constant. A stone is dropped from the top of a building. As the stone falls, it is uniformly accelerated, and its velocity increases uniformly in the downward direction. Consequently, the linear momentum of the stone also increases uniformly in the downward direction. According to Equation 29.8, the de Broglie wavelength of an object or particle is inversely proportional to the magnitude of its linear momentum. Therefore, as the stone falls, the de Broglie wavelength of the stone must decrease. _____________________________________________________________________________________________ 12. REASONING AND SOLUTION An electron and a neutron have different masses. According to Equation 29.8, EMBED "Equation" "Word Object8" \* mergeformat , the de Broglie wavelength of a particle is inversely proportional to the magnitude of its linear momentum. Therefore, if the electron and the neutron have different speeds such that the magnitudes of their respective momenta are the same, they will have the same de Broglie wavelength. _____________________________________________________________________________________________ 13. REASONING AND SOLUTION Suppose that in Figure 29.1, the electrons are replaced with protons that have the same speed. The speed of the protons is the same as that of the electrons; however, the mass of a proton is greater than the mass of an electron. Therefore, the magnitude of the linear momentum of a proton is greater than the linear momentum of an electron traveling at the same speed. According to Equation 29.8, EMBED "Equation" "Word Object8" \* mergeformat , the de Broglie wavelength of the "proton beam" will be smaller than that of the "electron beam." Equation 27.1 gives the condition that must be satisfied by the angle qð that locates the interference maxima in a Young's double-slit experiment: EMBED "Equation" "Word Object9" \* mergeformat , where m takes on integer values. According to Equation 27.1, the sine of the angle qð is directly proportional to the wavelength of the incident beam. Since the proton beam has a smaller wavelength than the electron beam, the values of sin qð that correspond to maxima will be smaller. Consequently, the difference in the corresponding values of qð will be smaller, and the angular separation between the bright fringes will decrease when the electrons are replaced by protons. _____________________________________________________________________________________________ PARTICLES AND WAVES Chapter 29 Conceptual Questions "#8•œ²!STfg˜™š›£¤ëîË(-C§¨ÙÚÛÜëì XZ…†˜™ÊËÌÍÕÖÕ27M„ûóíóçäàÞÞÙÖÎÅÙÞÞäàÙÂº±ÙÙÂ© ÙÞÞÙÖ˜ÙÞäàj¼*{7 EHøÿUj¼*{7 UVj½*{7 EHüÿUj½*{7 UVj¾*{7 EHüÿUj¾*{7 UVEHüÿj¿*{7 EHøÿUj¿*{7 UVEHøÿ jU656CJ 56CJ 56CJ(5CJ(OJQJ5CJ$8 !"#8–—½Ë)àà¾«›l[[[$„¸„Hþdð Æ`ú0ý¸ $dð Æ2`ú0ýhÐ p@à°€P ðÀ!$„Ð„0ýdð Æ`ú0ý„0ý„Ð„0ýdð Æ`ú0ý$„0ý„Ð„€ýdð Æ`ú0ý$!$$–P4Ö0°ÿ< @$dð$ Æ/`ú0ýÐ p@à°€P ðÀ! !"#8–—½Ë)*L M , - Õ34wxpÎÏ-‹ŒÛÆ$%„³´78ÍÎ&!'!¹!ñ!ò!d#e#$H$I$7%•%–% )þ)ÿ)š0V1X1å3C4D4Ô82939Q:C<¡<¢<f>Ä>Å>eFÃFÄFÅFßFàFG GGýýûù÷ù N)*L M , - Õ34wxpÎÏ-‹ŒÛÆ$%„³´78ÍÎîîîîîîîîîîîîîîîîîîîîîîîîîîî$„¸„Hþdð Æ`ú0ý¸„Œu y Ì Î ù ú >?@AIJ‡‹y"pÍÒè-Š¦(Æ#(>êë klžŸ ¡©ªÄÅ78jklmuv¨ýýýýøõíäøýýýýýáÝáÝýáÝøÚÒÉøøõÁ¸øýýøÚ°§øøÚj·*{7 EHöÿUj·*{7 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sPICTÿÿÿÿDECompObjÿÿÿÿÿÿÿÿÿÿÿÿRKObjInfo ÿÿÿÿTOlePres000ÿÿÿÿÿÿÿÿÿÿÿÿU(ub 2208 div 512 3 -1 roll exch div scale currentpoint translate 64 59 translate /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (7) -7 389 sh (9) 281 389 sh (10) 805 389 sh 384 /Times-Roman f1 (.) 185 389 sh 384 /Symbol f1 (\264) 547 389 sh 320 /Times-Roman f1 (14) 1176 217 sh 384 ns (Hz) 1645 389 sh 384 /Times-Roman f1 ( ) 1549 389 sh end MTsave restore ¿¡dIMATH= ˆ7‚.ˆ9†´ˆ1ˆ014Times | Hzsu ÿþÿÿÿÿÿÀFMathType Equationþÿÿÿ1ELO Equation€þÿÿÿÿÿÿÿ= ˆ7‚.ˆ9†´ˆ1ˆ014Times | HzOle10Native!ÿÿÿÿVAOle10FmtProgID ÿÿÿÿÿÿÿÿÿÿÿÿX _9308187485&ÀFàù>`½›>`½Ole ÿÿÿÿÿÿÿÿÿÿÿÿY EquationTžr¼èèžr ¢# . ÿÿÿ&ÿÿÿÿûôÿTimes-!E ûôÿPSymbol-ð!= ûôÿPIC #%ÿÿÿÿZTMETAÿÿÿÿÿÿÿÿÿÿÿÿ\LPICT$(ÿÿÿÿbmCompObjÿÿÿÿÿÿÿÿÿÿÿÿlKTimes-ð!hf &ÿÿÿÿ'ÿÿm#¡d·xpr Œ #"# ¾¡À currentpoint ¿" ¾,Times .ÿ+ E, Symbol) =) hf¡À¢/MTsave save def 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1120 div 448 3 -1 roll exch div scale currentpoint translate 64 58 translate /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (E) 6 262 sh (hf) 657 262 sh 384 /Symbol f1 (=) 347 262 sh end MTsave restore ¿¡d!MATH§ ƒE†=ƒhƒfcu ÿþÿÿÿÿÿÀFMathType Equationþÿÿÿ1ELO Equation€þÿÿÿÿÿÿÿObjInfo'*ÿÿÿÿnOlePres000ÿÿÿÿÿÿÿÿÿÿÿÿo(Ole10Native)+ÿÿÿÿpOle10FmtProgID ÿÿÿÿÿÿÿÿÿÿÿÿq ƒE†=ƒhƒf EquationLÒí¼èèm#¡d·xpr Œ #"# ¾¡À currentpoint ¿" ¾,Times .ÿ+ E, Symbol) =) hf¡À¢/MTsave save def 30 dict begin cur_930818746ÿÿÿÿÿÿÿÿ0ÀFÀÂ#>`½À,<>`½Ole ÿÿÿÿÿÿÿÿÿÿÿÿrPIC -/ÿÿÿÿsLPICTÿÿÿÿÿÿÿÿÿÿÿÿumrentpoint 3 -1 roll sub neg 3 1 roll sub 1120 div 448 3 -1 roll exch div scale currentpoint translate 64 58 translate /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (E) 6 262 sh (hf) 657 262 sh 384 /Symbol f1 (=) 347 262 sh end MTsave restore ¿¡d!MATH§ ƒE†=ƒhƒfcu ÿþÿÿÿÿÿÀFMathType Equationþÿÿÿ1ELO EqCompObj.2ÿÿÿÿKObjInfoÿÿÿÿÿÿÿÿÿÿÿÿOlePres00013ÿÿÿÿ‚(Ole10Nativeÿÿÿÿ4ÿÿÿÿƒþÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿ‡þÿÿÿ‰Š‹ŒŽ‘þÿÿÿ“þÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿšþÿÿÿœžŸ ¡¢£¤þÿÿÿ¦þÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿ¯°±²³´µ¶·¸¹ºþÿÿÿ¼þÿÿÿþÿÿÿþÿÿÿÀþÿÿÿþÿÿÿþÿÿÿÄþÿÿÿÆÇÈÉÊËÌÍÎþÿÿÿÐþÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿ×þÿÿÿÙÚÛÜÝÞßàáþÿÿÿãþÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿêþÿÿÿìíîïðñòóôþÿÿÿöþÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿýþÿÿÿÿuation€þÿÿÿÿÿÿÿ ƒE†=ƒhƒf EquationLÊ,èèOle10FmtProgID ÿÿÿÿÿÿÿÿÿÿÿÿ„ _930818745>,9ÀFÀ,<>`½€¾]>`½Ole ÿÿÿÿÿÿÿÿÿÿÿÿ…PIC 68ÿÿÿÿ†LPICTÿÿÿÿÿÿÿÿÿÿÿÿˆPCompObj7;ÿÿÿÿ’KObjInfoÿÿÿÿÿÿÿÿÿÿÿÿ”OlePres000:<ÿÿÿÿ•(P ¡d·xpr Œ " ¾¡À currentpoint ¿" ¾,Times .ÿ+ f +0¡À—/MTsave save def 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 416 div 480 3 -1 roll exch div scale currentpoint translate 64 58 translate /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (f) 56 262 sh 320 /Times-Roman f1 (0) 158 358 sh end MTsave restore ¿¡d$MATHõ ƒfˆ0re ÿþÿÿÿÿÿÀFMathType Equationþÿÿÿ1ELO Equation€þÿÿÿÿÿÿÿ ƒfˆ0 EquationOle10Nativeÿÿÿÿ=ÿÿÿÿ–Ole10FmtProgID ÿÿÿÿÿÿÿÿÿÿÿÿ— _930818744ÿÿÿÿÿÿÿÿBÀF€¾]>`½`‡n>`½Ole ÿÿÿÿÿÿÿÿÿÿÿÿ˜LÒí¼èèm#¡d·xpr Œ #"# ¾¡À currentpoint ¿" ¾,Times .ÿ+ E, Symbol) =) hf¡À¢/MTsave save def 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1120 div 448 3 -1 roll exch div scale currentpoint translate 64 58 translate /fs 0 def PIC ?Aÿÿÿÿ™LPICTÿÿÿÿÿÿÿÿÿÿÿÿ›mCompObj@Dÿÿÿÿ¥KObjInfoÿÿÿÿÿÿÿÿÿÿÿÿ§/cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (E) 6 262 sh (hf) 657 262 sh 384 /Symbol f1 (=) 347 262 sh end MTsave restore ¿¡d!MATH§ ƒE†=ƒhƒfcu ÿþÿÿÿÿÿÀFMathType Equationþÿÿÿ1ELO Equation€OlePres000CEÿÿÿÿ¨(Ole10NativeÿÿÿÿFÿÿÿÿ©Ole10FmtProgID ÿÿÿÿÿÿÿÿÿÿÿÿª _930818743k"KÀF`‡n>`½ >`½þÿÿÿÿÿÿÿ ƒE†=ƒhƒf EquationLýÌ,èè2W¡d·xpr Œ W"W ¾¡À currentpoint ¿" ¾,Times .ÿ+ KE +Ole ÿÿÿÿÿÿÿÿÿÿÿÿ«PIC HJÿÿÿÿ¬LPICTÿÿÿÿÿÿÿÿÿÿÿÿ®2CompObjIMÿÿÿÿ»Kmax, Symbol ( '=) hf)-) W + 0¡À/MTsave save def 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2784 div 480 3 -1 roll exch div scale currentpoint translate 64 58 translate /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (KE) -7 262 sh 320 ns (max) 527 358 sh 384 /Symbol f1 (=) 1210 262 sh (-) 1963 262 sh 384 /Times-Italic f1 (hf) 1520 262 sh (W) 2251 262 sh 320 /Times-Roman f1 (0) 2531 358 sh end MTsave restore ¿¡dIMATH= ÷ KEmax †=ƒhƒf†-ƒWˆ0su ÿþÿÿÿÿÿÀFMathType Equationþÿÿÿ1ELO Equation€þÿÿÿÿÿÿÿ= KEmax †=ƒhƒf†-ƒWˆ0ObjInfoÿÿÿÿÿÿÿÿÿÿÿÿ½OlePres000LNÿÿÿÿ¾(Ole10NativeÿÿÿÿOÿÿÿÿ¿AOle10FmtProgID ÿÿÿÿÿÿÿÿÿÿÿÿÁ EquationLWT,èèQ¡d·xpr Œ " ¾¡À currentpoint ¿" ¾,Times .ÿ+ W + 0¡À˜/MTsave save def 30 dict begin currentpoint 3 -1 roll s_930818742ÿÿÿÿÿÿÿÿTÀF >`½â >`½Ole ÿÿÿÿÿÿÿÿÿÿÿÿÂPIC QSÿÿÿÿÃLPICTÿÿÿÿÿÿÿÿÿÿÿÿÅQub neg 3 1 roll sub 544 div 480 3 -1 roll exch div scale currentpoint translate 64 59 translate /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (W) -13 261 sh 320 /Times-Roman f1 (0) 267 357 sh end MTsave restore ¿¡d$MATHæ ƒWˆ0re ÿþÿÿÿÿÿÀFMathType Equationþÿÿÿ1ELO EqCompObjRVÿÿÿÿÏKObjInfoÿÿÿÿÿÿÿÿÿÿÿÿÑOlePres000UWÿÿÿÿÒ(Ole10NativeÿÿÿÿXÿÿÿÿÓuation€þÿÿÿÿÿÿÿ ƒWˆ0 EquationLWT,èèOle10FmtProgID ÿÿÿÿÿÿÿÿÿÿÿÿÔ _930818741bP]ÀFÀ ª>`½`dÜ>`½Ole ÿÿÿÿÿÿÿÿÿÿÿÿÕPIC Z\ÿÿÿÿÖLPICTÿÿÿÿÿÿÿÿÿÿÿÿØQCompObj[_ÿÿÿÿâKObjInfoÿÿÿÿÿÿÿÿÿÿÿÿäOlePres000^`ÿÿÿÿå(Q¡d·xpr Œ " ¾¡À currentpoint ¿" ¾,Times .ÿ+ W + 0¡À˜/MTsave save def 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 544 div 480 3 -1 roll exch div scale currentpoint translate 64 59 translate /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (W) -13 261 sh 320 /Times-Roman f1 (0) 267 357 sh end MTsave restore ¿¡d$MATHæ ƒWˆ0re ÿþÿÿÿÿÿÀFMathType Equationþÿÿÿ1ELO Equation€þÿÿÿÿÿÿÿ ƒWˆ0 EquationOle10NativeÿÿÿÿaÿÿÿÿæOle10FmtProgID ÿÿÿÿÿÿÿÿÿÿÿÿç _930818739ÿÿÿÿÿÿÿÿfÀF€ä>`½ öý>`½Ole ÿÿÿÿÿÿÿÿÿÿÿÿèLä,èèN%¡d·xpr Œ %"% ¾¡À currentpoint ¿" ¾,Times .ÿ+ KE +max¡À/MTsave save def 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1184 div 480 3 -1 roll exch div scale currentpoint translate 64 59 translate /fs 0 def /cf 0 def /sf {exch duPIC ceÿÿÿÿéLPICTÿÿÿÿÿÿÿÿÿÿÿÿëNCompObjdhÿÿÿÿõKObjInfoÿÿÿÿÿÿÿÿÿÿÿÿ÷p /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (KE) -7 261 sh 320 ns (max) 527 357 sh end MTsave restore ¿¡d-MATH!¥ KEmax 3 ÿþÿÿÿÿÿÀFMathType Equationþÿÿÿ1ELO Equation€OlePres000giÿÿÿÿø(Ole10Nativeÿÿÿÿjÿÿÿÿù%Ole10FmtProgID ÿÿÿÿÿÿÿÿÿÿÿÿú _930818738YoÀF@—?`½@?`½þÿÿÿÿÿÿÿ! KEmax EquationLwhx €èèm †¡d·xpr Œ †" † ¾¡À currentpoint ¿" ¾,Times .ÿ+KE, SOle ÿÿÿÿÿÿÿÿÿÿÿÿûPIC lnÿÿÿÿüLPICTÿÿÿÿÿÿÿÿÿÿÿÿþmCompObjmqÿÿÿÿK þÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿ !"#$%&'()*+,-./0123456þÿÿÿ8þÿÿÿþÿÿÿþÿÿÿ<þÿÿÿþÿÿÿþÿÿÿ@þÿÿÿBCDEFGHIJþÿÿÿLþÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿSþÿÿÿUVWXYZ[\]^_`abcþÿÿÿeþÿÿÿþÿÿÿþÿÿÿiþÿÿÿþÿÿÿþÿÿÿmþÿÿÿopqrstuvwxyz{|}~€ymbol)= (1*2" ('mv (52 + =( I()mv)) (`2 (P2)m"H(j=( vp (}2 (t2)m"t¡À/MTsave save def 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4288 div 1024 3 -1 roll exch div scale currentpoint translate 64 42 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 8 th 925 548 moveto 207 0 rlineto stroke 16 th 2264 531 moveto 962 0 rlineto stroke 3652 531 moveto 529 0 rlineto stroke /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (KE) -7 630 sh 384 /Symbol f1 (=) 607 630 sh (=) 1946 630 sh (=) 3334 630 sh 320 /Times-Roman f1 (1) 948 412 sh (2) 948 859 sh (2) 1654 458 sh (2) 3010 217 sh (2) 3939 217 sh 384 ns (2) 2502 923 sh (2) 3673 923 sh 384 /Times-Italic f1 (mv) 1192 630 sh (mv) 2420 389 sh (m) 2710 923 sh (p) 3738 389 sh (m) 3881 923 sh 384 /Times-Roman f1 ($) 2278 389 sh ($) 2878 389 sh end MTsave restore ¿¡dMATHƒ KE†=ˆ1ˆ2 ƒmƒvˆ2 †=‚(ƒmƒv‚)ˆ2 ˆ2ƒm†=ƒpˆ2 ˆ2ƒm2 ÿþÿÿÿÿÿÀFMathType Equationþÿÿÿ1ELO Equation€þÿÿÿÿÿÿÿObjInfoÿÿÿÿÿÿÿÿÿÿÿÿOlePres000prÿÿÿÿ(Ole10Nativeÿÿÿÿsÿÿÿÿ‡Ole10FmtProgID ÿÿÿÿÿÿÿÿÿÿÿÿ ƒ KE†=ˆ1ˆ2 ƒmƒvˆ2 †=‚(ƒmƒv‚)ˆ2 ˆ2ƒm†=ƒpˆ2 ˆ2ƒm EquationLK{¬hèèÁo¡d·xpr Œ o"o ¾¡Àýÿÿÿ…ƒ„†‡ˆ‰Š‹ŒŽ‘’”—´•–˜™š›œŸž¡ ¢£¤¦¥¨§ª©«¬®¯°±²³¶µþÿÿÿ¸·º¹»¼½þÿÿÿ¾¿ÀÁÂÃÄþÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ_930818737ÿÿÿÿÿÿÿÿxÀF@?`½ Ê.?`½Ole ÿÿÿÿÿÿÿÿÿÿÿÿPIC uwÿÿÿÿLPICTÿÿÿÿÿÿÿÿÿÿÿÿÁ currentpoint ¿" ¾,Times .ÿ+p +max, Symbol (=)2)m) (KE +max (i)"'#"+ò#>¡Àe/MTsave save def 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3552 div 576 3 -1 roll exch div scale currentpoint translate 64 43 translate /thick 0 def /th { dup setlinewidth /thick exch def } def /sqr { 3 index div /thick exch def gsave translate dup dup neg scale dup 4 -1 roll exch div 3 1 roll div 0 setlinewidth newpath 0 0 moveto dup .34 mul 0 exch lineto .375 .214 rlineto dup thick add dup .375 exch lineto 2 index exch lineto dup thick 2 div sub dup 3 index exch lineto .6 exch lineto .375 0 lineto clip thick setlinewidth newpath dup .34 mul 0 exch moveto .15 .085 rlineto .375 0 lineto thick 2 div sub dup .6 exch lineto lineto stroke grestore } def 2226 472 384 1213 472 16 sqr /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (p) 28 341 sh (m) 1670 341 sh 320 /Times-Roman f1 (max) 212 437 sh 384 /Symbol f1 (=) 895 341 sh 384 /Times-Roman f1 (2) 1462 341 sh 384 /Times-Roman f1 ( $KE) 1947 341 sh ($) 3298 341 sh 320 ns (max) 2704 437 sh end MTsave restore ¿¡d_MATHS š ƒp‚m‚a‚x †= ˆ2ƒm (KEmax )ro ÿþÿÿÿÿÿÀFMathType Equationþÿÿÿ1ELO EqCompObjvzÿÿÿÿ7KObjInfoÿÿÿÿÿÿÿÿÿÿÿÿ9OlePres000y{ÿÿÿÿ:(Ole10Nativeÿÿÿÿ|ÿÿÿÿ;Wuation€þÿÿÿÿÿÿÿS ƒp‚m‚a‚x †= ˆ2ƒm (KEmax ) EquationL¸,èèOle10FmtProgID ÿÿÿÿÿÿÿÿÿÿÿÿ= _930818736†tÀFàñ7?`½À$a?`½Ole ÿÿÿÿÿÿÿÿÿÿÿÿ>PIC ~€ÿÿÿÿ?LZ¡d·xpr Œ " ¾¡À currentpoint ¿" ¾,Times .ÿ+ p +max¡À™/MTsave save def 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 864 div 480 3 -1 roll exch div scale currentpoint translate 64 59 translate /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /sh {moveto sPICTÿÿÿÿÿÿÿÿÿÿÿÿAZCompObjƒÿÿÿÿKKObjInfoÿÿÿÿÿÿÿÿÿÿÿÿMOlePres000‚„ÿÿÿÿN(how} def 384 /Times-Italic f1 (p) 28 261 sh 320 /Times-Roman f1 (max) 212 357 sh end MTsave restore ¿¡d*MATHG ƒp‚m‚a‚xin ÿþÿÿÿÿÿÀFMathType Equationþÿÿÿ1ELO Equation€þÿÿÿÿÿÿÿ ƒp‚m‚a‚xOle10Nativeÿÿÿÿ…ÿÿÿÿO"Ole10FmtProgID ÿÿÿÿÿÿÿÿÿÿÿÿP _930818735ÿÿÿÿÿÿÿÿŠÀFÀ$a?`½€¶‚?`½Ole ÿÿÿÿÿÿÿÿÿÿÿÿQ EquationL E(lèèÐB¡d·xpr Œ B"B ¾¡À currentpoint ¿" ¾, Symbol .ÿ+l,Times +min (=( 0h((p +max" &¡À£/MTsave save def 30 dict begin currentpoint 3 PIC ‡‰ÿÿÿÿRLPICTÿÿÿÿÿÿÿÿÿÿÿÿTÐCompObjˆŒÿÿÿÿdKObjInfoÿÿÿÿÿÿÿÿÿÿÿÿf-1 roll sub neg 3 1 roll sub 2112 div 992 3 -1 roll exch div scale currentpoint translate 64 41 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1162 404 moveto 851 0 rlineto stroke /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (l) -9 503 sh 320 /Times-Roman f1 (min) 212 600 sh (max) 1406 892 sh 384 /Symbol f1 (=) 844 503 sh 384 /Times-Italic f1 (h) 1493 262 sh (p) 1222 796 sh end MTsave restore ¿¡dOMATHC H „l‚m‚i‚n †=ƒhƒp‚m‚a‚x12 ÿþÿÿÿÿÿÀFMathType Equationþÿÿÿ1ELO Equation€þÿÿÿÿÿÿÿOlePres000‹ÿÿÿÿg(Ole10NativeÿÿÿÿŽÿÿÿÿhGOle10FmtProgID ÿÿÿÿÿÿÿÿÿÿÿÿj _930818734¡}“ÀF`“?`½p?`½C „l‚m‚i‚n †=ƒhƒp‚m‚a‚x EquationL04P @èè#„¡d·xpr Œ „"„ ¾¡À currentpoint ¿" ¾, Symbol .ÿ+l,Ole ÿÿÿÿÿÿÿÿÿÿÿÿkPIC ’ÿÿÿÿlLPICTÿÿÿÿÿÿÿÿÿÿÿÿn#CompObj‘•ÿÿÿÿƒK‚þÿÿÿ„þÿÿÿþÿÿÿþÿÿÿˆþÿÿÿþÿÿÿþÿÿÿŒþÿÿÿŽ‘’“”•–þÿÿÿ˜þÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿŸþÿÿÿ¡¢£¤¥¦§¨©ªþÿÿÿ¬þÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿ³þÿÿÿµ¶·¸¹º»¼½þÿÿÿ¿þÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿÆþÿÿÿÈÉÊËÌÍÎÏÐÑÒþÿÿÿÔþÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿÛþÿÿÿÝÞßàáâãäåæçþÿÿÿéþÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿðþÿÿÿòóôõö÷øùúûüþÿÿÿþþÿÿÿþÿÿÿTimes)')-)l) =) h)/()mc))('[)&])()1)-) cos )q))¡À§/MTsave save def 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4224 div 512 3 -1 roll exch div scale currentpoint translate 64 56 translate /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (l) -9 296 sh (l) 583 296 sh (q) 3768 296 sh 384 /Times-Roman f1 (\251) 205 296 sh (/$) 1584 296 sh ($) 2289 296 sh ($) 2522 296 sh 384 /Symbol f1 (-) 357 296 sh (=) 914 296 sh /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1247 /Symbol f3 ([) 1199 321 sh (]) 2416 321 sh 384 /Symbol f1 (-) 2869 296 sh 384 /Times-Italic f1 (h) 1335 296 sh (mc) 1832 296 sh 384 /Times-Roman f1 (1) 2626 296 sh 384 /Times-Roman f1 (cos ) 3161 296 sh ($) 4007 296 sh end MTsave restore ¿¡d^MATHR½ „l‚©†-„l†=ƒh‚/‚(ƒmƒc‚)–[–]‚(ˆ1†-cos „q) r ÿþÿÿÿÿÿÀFMathType Equationþÿÿÿ1ELO Equation€þÿÿÿÿÿÿÿR „l‚©†-„l†=ƒh‚/‚(ƒmƒc‚)–[–]‚(ˆ1ObjInfoÿÿÿÿÿÿÿÿÿÿÿÿ…OlePres000”–ÿÿÿÿ†(Ole10Nativeÿÿÿÿ—ÿÿÿÿ‡VOle10FmtProgID ÿÿÿÿÿÿÿÿÿÿÿÿ‰ †-cos „q) EquationLöíÐèè€$¡d·xpr Œ $"$ ¾¡À currentpoint ¿" ¾,Times .ÿ+ h)/()mc))¡Àº/MTsave save def 30 dict begin currentpoint 3_930818732ÿÿÿÿÿÿÿÿœÀF µ?`½à8¾?`½Ole ÿÿÿÿÿÿÿÿÿÿÿÿŠPIC ™›ÿÿÿÿ‹LPICTÿÿÿÿÿÿÿÿÿÿÿÿ€ -1 roll sub neg 3 1 roll sub 1152 div 448 3 -1 roll exch div scale currentpoint translate 64 58 translate /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (h) -8 262 sh (mc) 489 262 sh 384 /Times-Roman f1 (/$) 241 262 sh ($) 946 262 sh end MTsave restore ¿¡d'MATH ƒh‚/‚(ƒmƒc‚)tp ÿþÿÿÿÿÿÀFMathType Equationþÿÿÿ1ELO EqCompObjšžÿÿÿÿ—KObjInfoÿÿÿÿÿÿÿÿÿÿÿÿ™OlePres000Ÿÿÿÿÿš(Ole10Nativeÿÿÿÿ ÿÿÿÿ›uation€þÿÿÿÿÿÿÿ ƒh‚/‚(ƒmƒc‚) EquationLìí\èèOle10FmtProgID ÿÿÿÿÿÿÿÿÿÿÿÿœ _930818731³˜¥ÀFÚÅ?`½€“ð?`½Ole ÿÿÿÿÿÿÿÿÿÿÿÿPIC ¢¤ÿÿÿÿžLPICTÿÿÿÿÿÿÿÿÿÿÿÿ ®CompObj£§ÿÿÿÿ«KObjInfoÿÿÿÿÿÿÿÿÿÿÿÿOlePres000¦¨ÿÿÿÿ®(®+¡d·xpr Œ +"+ ¾¡À currentpoint ¿" ¾,Times .ÿ+ p, Symbol) =) E) /)c¡ÀÕ/MTsave save def 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1376 div 448 3 -1 roll exch div scale currentpoint translate 64 59 translate /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (p) 28 261 sh (E) 638 261 sh (c) 1106 261 sh 384 /Symbol f1 (=) 314 261 sh 384 /Times-Roman f1 (/) 939 261 sh end MTsave restore ¿¡d$MATH_ ƒp†=ƒE‚/ƒcre ÿþÿÿÿÿÿÀFMathType Equationþÿÿÿ1ELO Equation€þÿÿÿÿÿÿÿ ƒp†=ƒE‚/ƒcOle10Nativeÿÿÿÿ©ÿÿÿÿ¯Ole10FmtProgID ÿÿÿÿÿÿÿÿÿÿÿÿ° _930818730ÿÿÿÿÿÿÿÿ®ÀF 4ø?`½`\@`½Ole ÿÿÿÿÿÿÿÿÿÿÿÿ± EquationLöÊÐèèm $¡d·xpr Œ $" $ ¾¡À currentpoint ¿" ¾,Times .ÿ+ E, Symbol) =)pc¡À¢/MTsave save def 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1152 div 416 3 -1 roll PIC «ÿÿÿÿ²LPICTÿÿÿÿÿÿÿÿÿÿÿÿ´mCompObj¬°ÿÿÿÿ¾KObjInfoÿÿÿÿÿÿÿÿÿÿÿÿÀexch div scale currentpoint translate 64 27 translate /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (E) 6 261 sh (pc) 693 261 sh 384 /Symbol f1 (=) 347 261 sh end MTsave restore ¿¡d!MATH¬ ƒE†=ƒpƒccu ÿþÿÿÿÿÿÀFMathType Equationþÿÿÿ1ELO Equation€þÿÿÿÿÿÿÿ ƒE†=ƒpƒc EquationLìí\èèÿ+¡d·xpr Œ +"+ ¾¡ÀOlePres000¯±ÿÿÿÿÁ(Ole10Nativeÿÿÿÿ²ÿÿÿÿÂOle10FmtProgID ÿÿÿÿÿÿÿÿÿÿÿÿÃ _930818729Åª·ÀF€ý@`½@*@`½Ole ÿÿÿÿÿÿÿÿÿÿÿÿÄPIC ´¶ÿÿÿÿÅLPICTÿÿÿÿÿÿÿÿÿÿÿÿÇÿCompObjµ¹ÿÿÿÿÓK currentpoint ¿" ¾, Symbol .ÿ+ l)=,Times) h)/)p¡À)/MTsave save def 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1376 div 448 3 -1 roll exch div scale currentpoint translate 64 36 translate /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (l) -9 284 sh 384 /Symbol f1 (=) 322 284 sh 384 /Times-Italic f1 (h) 632 284 sh (p) 1088 284 sh 384 /Times-Roman f1 (/) 881 284 sh end MTsave restore ¿¡d$MATH| „l†=ƒh‚/ƒpre ÿþÿÿÿÿÿÀFMathType Equationþÿÿÿ1ELO Equation€þÿÿÿÿÿÿÿ „l†=ƒh‚/ƒpObjInfoÿÿÿÿÿÿÿÿÿÿÿÿÕOlePres000¸ºÿÿÿÿÖ(Ole10Nativeÿÿÿÿ»ÿÿÿÿ×Ole10FmtProgID ÿÿÿÿÿÿÿÿÿÿÿÿØ EquationLìí\èèÿ+¡d·xpr Œ +"+ ¾¡À currentpoint ¿" ¾, Symbol .ÿ+ l)=,Times) h)/)p¡À)/MTsave save def 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1376 div 448 3 _930818728ÿÿÿÿÿÿÿÿÀÀF·3@`½ÀHU@`½Ole ÿÿÿÿÿÿÿÿÿÿÿÿÙPIC ½¿ÿÿÿÿÚLPICTÿÿÿÿÿÿÿÿÿÿÿÿÜÿ-1 roll exch div scale currentpoint translate 64 36 translate /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (l) -9 284 sh 384 /Symbol f1 (=) 322 284 sh 384 /Times-Italic f1 (h) 632 284 sh (p) 1088 284 sh 384 /Times-Roman f1 (/) 881 284 sh end MTsave restore ¿¡d$MATH| „l†=ƒh‚/ƒpre ÿCompObj¾ÂÿÿÿÿèKObjInfoÿÿÿÿÿÿÿÿÿÿÿÿêOlePres000ÁÃÿÿÿÿë(Ole10NativeÿÿÿÿÄÿÿÿÿìþÿÿÿÿÿÀFMathType Equationþÿÿÿ1ELO Equation€þÿÿÿÿÿÿÿ „l†=ƒh‚/ƒp EquationLìí\èèOle10FmtProgID ÿÿÿÿÿÿÿÿÿÿÿÿí _930818727Î¼ÉÀFÀHU@`½ f@`½Ole ÿÿÿÿÿÿÿÿÿÿÿÿîPIC ÆÈÿÿÿÿïLÿ+¡d·xpr Œ +"+ ¾¡À currentpoint ¿" ¾, Symbol .ÿ+ l)=,Times) h)/)p¡À)/MTsave save def 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1376 div 448 3 -1 roll exch div scale currentpoint translate 64 36 translate /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /PICTÿÿÿÿÿÿÿÿÿÿÿÿñÿCompObjÇËÿÿÿÿýKObjInfoÿÿÿÿÿÿÿÿÿÿÿÿÿOlePres000ÊÌÿÿÿÿ(ns {cf sf} def /sh {moveto show} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (l) -9 284 sh 384 /Symbol f1 (=) 322 284 sh 384 /Times-Italic f1 (h) 632 284 sh (p) 1088 284 sh 384 /Times-Roman f1 (/) 881 284 sh end MTsave restore ¿¡d$MATH| „l†=ƒh‚/ƒpre ÿþÿÿÿÿÿÀFMathType Equationþÿÿÿ1ELO Equation€þÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿ þÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿþÿÿÿ þÿÿÿ"#$%&'(þÿÿÿ*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\þÿÿÿ^þÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿþÿÿÿÿÿÿÿ „l†=ƒh‚/ƒp EquationL; Ê<èèB C¡d·xpr Œ C" C ¾¡À currentpoint ¿" ¾,Times .ÿ+ sin, Ole10NativeÿÿÿÿÍÿÿÿÿOle10FmtProgID ÿÿÿÿÿÿÿÿÿÿÿÿ _930818724ÿÿÿÿÿÿÿÿÒÀFÀ²m@`½€D@`½Ole ÿÿÿÿÿÿÿÿÿÿÿÿPIC ÏÑÿÿÿÿLPICTÿÿÿÿÿÿÿÿÿÿÿÿBCompObjÐÔÿÿÿÿKObjInfoÿÿÿÿÿÿÿÿÿÿÿÿSymbol)q) =) m)l) /)d¡ÀM/MTsave save def 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2144 div 416 3 -1 roll exch div scale currentpoint translate 64 36 translate /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (sin) -19 284 sh (/) 1660 284 sh /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (q) 450 284 sh (l) 1369 284 sh 384 /Symbol f1 (=) 778 284 sh 384 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