1. Section 2.6
Related Rates
Read Guidelines For Solving Related Rates Problems on p. 150.

2. Example 1
A 17 foot ladder is sliding down a wall. The base of the ladder is moving away from the wall at a rate of 2 feet per second. How fast is the top of the ladder moving down the wall when the base of the ladder is 8 feet away from the wall?

3. Step 1: Draw a sketch and label known and unknown quantities
Step 1: Draw a sketch and label known and unknown quantities. Write down what is given and what is to be determined.
s = 17 ft. y x

4. Step 2: Write an equation involving the variables whose rates of changes either are given or to be determined. Step 3: Differentiate each side with respect to t.

5. Step 4: Substitute all known values for the variables and their rates of change. Then solve for the required rate of change. The top of the ladder is moving down the wall at a rate of about −1.067 ft/s when the base of the ladder is 8 ft. from the wall.

6. Example 2
An adventurer rides down a zip-line at a speed of 80 mph. If the angle of depression of the zip-line is 75°, how fast is the zip-liner’s altitude changing?
75° z h

7. 75° z h
The adventurer’s altitude is decreasing by a rate of about mph when the angle of depression is 75°.

8. Example 3
A 6 foot tall man walks away from a 22 foot street light at a speed of 8 feet per second. What is the rate of change of the length of his shadow when he is 19 feet away from the light? Also, at what rate is the tip of his shadow moving?

9. 22 6 x s

10. The length of the man’s shadow is increasing at a rate of 3 ft/s.22 6 x s
The length of the man’s shadow is increasing at a rate of 3 ft/s.

11. 22 6 x s
y = x + s
The tip of his shadow is moving at rate of 11 ft/s when he is 19 ft. from the street light.

12. Example 4
A large spherical balloon is being inflated and its volume is increasing at a rate of 3.5 cubic feet per minute. What is the rate of change of the radius when the radius is 7 feet?

13. r

14. r
The radius of the spherical balloon is increasing at a rate of about ft/sec when the radius is 7 ft.

15. Example 5
An upside-down conical tank full of water has a “base” radius of 3 meters and a height of 5 meters. The is being drained at a rate of 2 cubic meters per meter. What is the rate of change of the height of the water when the height is 4 meters?

16. 3 r 5 h

17. 3 r 5 h 3 5 r h

18. 3 r 5 h
The height of the water is decreasing at a rate of about meter/min when its height is 4 meters.

19. The Grand Finale!!!!
Example 6

20. An upside-down conical tank full of water has a “base” radius of 5 feet and a height of 7 feet. The water is being drained into a cylindrical tank with radius of 5 feet and height 6 feet. The radius of the water in the conical tank is decreasing at a rate of 2 feet per minute. At what rate does the water level in the cylindrical tank rise when the water level in the conical tank is 3 feet?

21. 5 r 7 h1 5 5 6 h2

22. 5 r 5 7 r h1 7 h1 5 5 6 h2

23. 5 r 7 h1 5 5 6 h2

24. 5 r 7 h1 5 5 6 h2

25. The water level in the cylindrical tank is increasing at rate of about ft./min when the water level of the conical tank is 3 ft.