1. Related Rates
AP Calculus
Keeper 26

2. Related Rates Organization
Read the problem
Answer these questions:
Know: What rate are you given? (ie. 𝑑𝑣 𝑑𝑡 )
Find: What rate are you asked to find?
When: Most of the time you are given a time when you want to find the rate. This information is only sued after the derivative is taken. If this information is not given, then you will not need it.
Write an equation that relates the Know rate with the Find rate.

3. Related Rates Organization Cont.
Take a derivative. Since we are concerned about When things occur we will be taking derivative with respect to time (𝑡) and we will need to do implicit differentiation.
Substitute in the known rate and the when time.
Evaluate and label your answer.

4. Sphere Problem
Joe inflates a spherical balloon. Air is entering the balloon at a rate of 15 𝑐 𝑚 3 𝑠𝑒𝑐 . How fast is the radius changing when the radius is 10 𝑐𝑚.

5. Pebble Problem
A pebble is thrown into a pond forming ripples whose radius increases at the rate of 4 in/sec. How fast is the area of the ripple changing when the radius is one foot?

6. Example
The radius of a circle is increasing at the rate of 2 in/sec. At what rate is the area increasing when the circumference of the circle is 12𝜋 in.?

7. Cotton Doily Problem
A circular cotton doily with radius 22 cm is inadvertently thrown in the dryer and starts shrinking so that the radius is decreasing at a rate of 2 cm/min. At what rate is the area enclosed by the circle decreasing 5 minutes after the doily is thrown the dryer?

8. Ice Problem
A piece of ice cut in the shape of a cube melts uniformly so that its volume decreases at 3 𝑐 𝑚 3 /sec. How fast is the surface area decreasing when the edge of the cube is 5 cm.

9. Balloon Problem
A spherical balloon is filled with air at the rate of 8 𝑖 𝑛 3 /sec. How fast is the diameter of the balloon increasing when the volume of the balloon is 36𝜋 𝑖 𝑛 3 ?

10. Ladder Problem
A 20 foot ladder is leaning against a house. The foot of the ladder begins to slide away from the house at a rate of 2 feet/second. How fast is the top of the adder sliding down the wall when the foot of the ladder is 12 feet from the house?

11. Example
The length 𝑙 of a rectangle is decreasing at the rate of 2 cm/sec while the width 𝑤 is increasing at the rate of 2 cm/sec. When 𝑙=12 cm and 𝑤=5 𝑐𝑚, find the rates of change in (a) the area (b) the perimeter and (𝑐) the length of the diagonal of the rectangle.

12. Example
A ladder 15 m tall slides down the side of a water tower. When the bottom end is 11m from the tower, the opposite end is sliding down at a rate of 3m/h
At the instant, how fast is the bottom of the ladder moving away from the tower?
How fast is the area of the region created between the ladder, the ground, and the tower changing?
2.78 m/h
𝑚 2 /ℎ𝑟

13. Example
Darth Vader’s spaceship is approaching the origin along the positive y axis at 50 km/sec. Meanwhile, his daughter Ella’s spaceship is moving away from the origin along the positive x-axis at 8- km/sec. When Darth is at 𝑦= 1200 𝑘𝑚 and Ella is at 𝑥=500 𝑘𝑚, is the distance between them increasing or decreasing? At what rate?

14. Example
A winch at the end of the dock is 9 ft above the level of the deck of a boat. A rope attached to the deck is being hauled in by the winch at a rate of 3 ft/sec. How fast is the boat being pulled toward the dock when 15 ft of rope are out?

15. Example
An angler has hooked a fish. The fish was swimming is an east-west direction along a line 40 ft north of the angler. If the line is leaving the reel at a rate od 7 ft/sec when the fish is 60 ft from the angler, how fast is the fish traveling?

16. Example
All edges of a cube are expanding at a rate of 3 cm/sec. How fast is the volume changing when each edge is 1 cm.

17. Example
The volume of a cube is decreasing at the rate of 12 cubic meters per hour. How fast is the total surface area decreasing when the surface area is 24 𝑚 2 ?

18. Example
A spherical balloon is inflates at the rate of four cubic feet per minute. At what rate is the radius changing when 𝑟=24 𝑖𝑛?

19. Example
A rectangle has a fixed area of 100 𝑢𝑛𝑖𝑡 𝑠 2 and its length L in increasing at 2 units/sec. Find the length L at the instant when the width is decreasing at 0.5 units/sec.

20. Example
A screen saver displays the outline of a 3 cm by 2 cm rectangle and then expands the rectangle at the rate of 4 cm/sec and the proportions of the rectangle never change. How fast is the area of the rectangle increasing when its dimensions are 12 cm by 8 cm?

21. Example
A ladder leans against a wall with the bottom of the ladder 8 ft from the wall. The top of the ladder slips down the wall at a rate of 4 ft/sec while the bottom of the ladder is being pulled away at a rate of 3 ft/sec. How long is the ladder?

22. Example
A boat is pulled toward a pier by means of a taut cable. If the boat is 20 ft below the level of the pier and the cable is pulled in at a rate of 36 ft/min, how fast is the boat moving when it is 48 ft from the base of the pier?

23. Example
Two vehicles are approaching an intersection, one truck from the west at 15 m/sec and one van from the north at 20 m/sec. How fast is the distance between the vehicles changing at the instant the truck is 60m west and the van 80m north of the intersection?

24. Example
Car A is going west at 50 mph and car B is headed north at 60 mph. Both are headed for the intersection of the two roads. At what rate are the cars approaching each other when car A is 0.3 mi and car B is 0.4 mi from the intersection?

25. Shadow Problem
A pickpocket walking away from a 10 meter tall lamppost is 2 meters tall. He walks at a rate of 1.5 m/sec. How fast is his shadow growing when he is 5 meter from the lamppost?

26. Cone Problem
A water tank is in the shape of an inverted cone with a diameter of 12 feet and a depth of 8 feet. If the water is 3 feet deep and is rising at a rate of 2 feet/hr, at what rate is the volume changing?