1. Related Rates
AP Calculus AB

2. In related-rates problems, the goal is to calculate an unknown rate of change in terms of other rates of change that are known.
The “sliding ladder problem” is a good example: A ladder leans against a wall as the bottom is pulled away at constant velocity. How fast does the top of the ladder move?

3. Process for related rates
Draw a picture. Label constant values and assign variables to things that change.
Translate the given information in the problem into “calculus-speak.” Do the same thing for what you are asked to find. For example, the rate of change of the area is ft 2 min becomes 𝑑𝐴 𝑑𝑡 =15 ft 2 min .
Write a formula/equation related the variables whose rates of change you seek and the variables whose rates of change you are given. IMPORTANT: AT THIS STAGE YOU MAY SUBSTITUTE FOR A QUANTITY THAT IS CONSTANT; HOWEVER, DON’T FREEZE YOUR PROBLEM BY SUBSTITUTING A NUMBER FOR A QUANTITY THAT IS CHANGING – KEEP VARIABLES VARIABLES!
Differentiate implicitly with respect to time. Use all differentiation rules that apply.
Now you can plug in numbers and do calculations.
Use complete sentences to answer the question that is asked.

4. Translate into calculus notation
The area of a circle is increasing at a rate of 6 square inches per minute.
𝑑𝐴 𝑑𝑡 =6 in 2 min
The volume of a cone is decreasing at a rate of 2 cubic feet per second.
𝑑𝑉 𝑑𝑡 =−2 ft 3 sec
The population of a city is growing at a rate of 3 people per day.
𝑑𝑃 𝑑𝑡 =3 people day

5. Translate into words 𝑑𝐶 𝑑𝑡 =2 in sec
The circumference of a circle is increasing at a rate of 2 inches per second.
𝑑𝑟 𝑑𝑡 =40 cm sec
The radius of a circle is increasing at a rate of 40 centimeters per second.
𝑑𝑉 𝑑𝑡 =−28 ft 3 min
The volume of a cube is decreasing at a rate of 28 cubic feet per minute.

6. Differentiate Differentiate with respect to t. 𝐴=𝜋 𝑟 2
𝑑𝐴 𝑑𝑡 =2𝜋𝑟 𝑑𝑟 𝑑𝑡
𝑎 2 + 𝑏 2 = 𝑐 2
2𝑎 𝑑𝑎 𝑑𝑡 +2𝑏 𝑑𝑏 𝑑𝑡 =2𝑐 𝑑𝑐 𝑑𝑡
𝐴= 1 2 𝑏ℎ
𝑑𝐴 𝑑𝑡 = 1 2 𝑏 𝑑ℎ 𝑑𝑡 + ℎ 𝑑𝑏 𝑑𝑡
𝑑𝐴 𝑑𝑡 = 1 2 𝑏 𝑑ℎ 𝑑𝑡 ℎ 𝑑𝑏 𝑑𝑡

7. Example
The radius of a circle is increasing at a constant rate of 0.4 meters per second. What is the rate of increase in the area of the circle at the instant when the circumference is 60𝜋?

8. Example
A 15 foot ladder is sliding down a building at a constant rate of 2 feet per minute. How fast is the base of the ladder moving away from the building when the base of the ladder is 9 feet from the building?

9. Example
Water pours into a fish tank at a rate of 0.3 cubic meters per minute. How fast is the water level rising if the base of the tank is a rectangle of dimensions 2 x 3 meters?

10. Example
Water pours into a conical tank of height 10 meters and radius 4 meters at a rate of 6 cubic meters per minute. At what rate is the water level rising when the level is 5 meters high?
V=1/3pihr2

11. 2002 FRQ #5
A container has the shape of an open right circular cone, as shown in
the figure above. The height of the container is 10 cm and the diameter
of the opening is 10 cm. Water in the container is evaporating so that
its depth h is changing at the constant rate of −3 10 cm/hr. (Note:
The volume of a cone of height h and radius r is given by 𝑉= 1 3 𝜋 𝑟 2 ℎ.)
Find the volume V of water in the container when h = 5 cm Indicate units of measure.
Find the rate of change of the volume of water in the container, with respect to time, when h = 5 cm. Indicate units of measure.
Show that the rate of change of the volume of water in the container due to evaporation is directly proportional to the exposed surface area of the water. What is the constant of proportionality?

12. 2008 FRQ #3
Oil is leaking from a pipeline on the surface of a lake and forms an oil slick whose volume increases at a constant rate of 2000 cubic centimeters per minute. The oil slick takes the form of a right circular cylinder with both its radius and height changing with time. (Note: The volume V of a right circular cylinder with radius r and height h is given by 𝑉=𝜋 𝑟 2 ℎ.)
At the instant when the radius of the oil slick is 100 centimeters and the height is 0.5 centimeter, the radius is increasing at the rate of 2.5 centimeters per minute. At this instant, what is the rate of change of the height of the oil slick with respect to time, in centimeters per minute?
A recovery device arrives on the scene and begins removing oil. The rate at which oil is removed is 𝑅 𝑡 =400 𝑡 cubic centimeters per minute, where t is the time in minutes since the device began working. Oil continues to leak at the rate of 2000 cubic centimeters per minute. Find the time t when the oil slick reaches its maximum volume. Justify your answer.
By the time the recovery device began removing oil, 60,000 cubic centimeters of oil had already leaked. Write, but do not evaluate, an expression involving an integral that gives the volume of oil at the time found in part (b).

13. Example
A spy uses a telescope to track a rocket launched vertically from a launching pad 6 kilometers away. At a certain moment, the angle θ between the telescope and the ground is equal to 𝜋 3 and is changing at a rate of 0.9 radians per minute. What is the rocket’s velocity at that moment?

14. In Class
Page
Numbers 1-17

15. HW
Page 200
Numbers 18-28