1. AP Calculus AB
5.6 Related Rates

2. First, a review problem:
Consider a sphere of radius 10cm.
If the radius changes 0.1cm (a very small amount) how much does the volume change?
The volume would change by approximately

3. Now, suppose that the radius is changing at an instantaneous rate of 0
Now, suppose that the radius is changing at an instantaneous rate of 0.1 cm/sec.
(Possible if the sphere is a soap bubble or a balloon.)
The sphere is growing at a rate of
Note: This is an exact answer, not an approximation like
we got with the differential problems.

4. Water is draining from a cylindrical tank at 3 liters/second
Water is draining from a cylindrical tank at 3 liters/second. How fast is the surface dropping?
Find
(We need a formula to relate V and h. )
(r is a constant.)

5. Steps for Related Rates Problems:
1. Draw a picture (sketch).
2. Write down known information.
3. Write down what you are looking for.
4. Write an equation to relate the variables.
5. Differentiate both sides with respect to t.
6. Evaluate.

6. Truck A travels east at 40 mi/hr. Truck B travels north at 30 mi/hr.
Truck Problem:
Truck A travels east at 40 mi/hr.
Truck B travels north at 30 mi/hr.
How fast is the distance between the trucks changing 6 minutes later? B A

7. Truck A travels east at 40 mi/hr. Truck B travels north at 30 mi/hr.
Truck Problem:
Truck A travels east at 40 mi/hr.
Truck B travels north at 30 mi/hr.
How fast is the distance between the trucks changing 6 minutes later? B A

8. Draining Conical Reservoir Water is flowing at the rate of 50 m3/min from a concrete conical reservoir (vertex down) of base radius 45 m and height 6 m. How fast is the water level falling when the water is 5 m deep? How fast is the radius of the water’s surface changing at that moment?
45 m
6 m

9. Draining Conical Reservoir Water is flowing at the rate of 50 m3/min from a concrete conical reservoir (vertex down) of base radius 45 m and height 6 m. How fast is the water level falling when the water is 5 m deep? How fast is the radius of the water’s surface changing at that moment?

10. Draining Conical Reservoir Water is flowing at the rate of 50 m3/min from a concrete conical reservoir (vertex down) of base radius 45 m and height 6 m. How fast is the water level falling when the water is 5 m deep? How fast is the radius of the water’s surface changing at that moment?

11. Example 2: Hot Air Balloon Problem
Given:
How fast is the balloon rising?
Find

12. Hot Air Balloon Problem:
Given:
How fast is the balloon rising?
Find

13. Sliding Ladder. A 13-ft ladder is leaning up against a house when its base starts to slide away. By the time the base is 12-ft from the house, the base is moving at the rate of 5 ft/sec.
How fast is the ladder sliding down the wall at that moment?
At what rate is the area of the triangle formed by the ladder, wall, and ground changing at that moment?
At what rate is the angle θ between the ladder and the ground changing at that moment?

14. (a) How fast is the ladder sliding down the wall at that moment?
Substitute the known values.

15. (b) At what rate is the area of the triangle formed by the ladder, wall, and ground changing at that moment?

16. (c) At what rate is the angle θ between the ladder and the ground changing at that moment?p