Lesson Plan #6
Lesson Plan #19
Class: AP Calculus Date: Wednesday October 20th, 2010
Topic: Related Rates Aim: How do we find related rates?
Objectives:
1) Students will be able to find related rates
HW# 19:
The formula for the volume of a cone is [pic]. Find the rate of change of the volume if [pic]is 2 inches per minute and [pic], when
A) [pic] B) [pic]
Do Now:
Assume both [pic]and [pic]are differentiable functions of [pic]. Find [pic]if [pic]and [pic]=3 if [pic]
Write the Aim and Do Now
Get students working!
Take attendance
Give back work
Go over the HW
Collect HW
Go over the Do Now
Example #1:
Assume both [pic]and [pic]are differentiable functions of [pic]. Find [pic]when [pic], and [pic]if [pic]
Example #2:
Assume both [pic]and [pic]are differentiable functions of [pic]. Find [pic]when [pic]and [pic]if [pic]
Now let’s take a look at some application problems that involve two or more quantities related to time.
Example #1:
The radius [pic]of a circle is increasing at a rate of [pic]centimeters per minute. Find the rate of change of the area of the circle when the radius is [pic]centimeters.
Example #2:
The radius [pic]of a sphere is increasing at a rate of 2 inches per minute. Find the rate of change of Volume of the sphere when the radius = [pic]inches.
Example #3:
All edges of a cube are expanding at a rate of 3 centimeters per second. How fast is the volume changing when each edge is 1 centimeter.
Example #4:
An Airplane is flying on a flight path that will take it directly over a radar tracking station. If the distance between the plane and the radar traction station ([pic]) and the plane is decreasing at a rate of 400 mph at the moment that distance is 10 miles, what is the speed of the plane, to the nearest mile per hour, if the plane is flying at an altitude of 5 miles?
On Your Own:
1) A spherical balloon is inflated with gas at the rate of 500 cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is
30 centimeters
2) The formula for the volume of a cone is[pic]. Find the rate of change of volume if [pic]is 2 inches per minute and [pic] when [pic]inches.
3) At a sand gravel plant, sand is falling off a conveyor and onto a conical pile at a rate of 10 cubic feet per minute. The diameter of the base of the cone is approximately 3 times the altitude. At what rate is the height of the pile changing when the pile is 15 feet high?
4) A ladder 25 long is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of 2 feet per second. How fast is the top moving down the wall when the base of the ladder is 7 feet from the wall? Consider the triangle formed by the side of the house, the ladder and the ground. Find the rate at which the area of the triangle is changing when the base of the ladder is 7 feet from the wall.
If Enough Time:
1) If one leg of a right triangle AB increases at the rate of 2 inches per second, while the other leg AC decreases at the rate of 3 inches per second, find how fast the hypotenuse is changing when AB = 6 feet and AC = 8 feet.
2) The diameter and height of a paper cup in the shape of a cone are both 4 inches, and water is leaking out at a rate of [pic]cubic inch per second. Find the rate at which the water level is dropping when the diameter of the surface is 2 inches.
3) A balloon is being filled with helium at the rate of 4ft[pic]/min. The rate, in square feet per minute at which the surface area is increasing when the volume is [pic]ft[pic] is
A) [pic] B) 2 C) 4 D) 1 E) [pic]
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