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1. Consider a sphere of radius 10cm. Warmup: If the radius changes 0.1cm (a very small amount) how much does the volume change?

2. The sphere is growing at a rate of . Now back to our volume problem, 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.)

3. Related Rates

4. r A = p r 2 Suppose a circle is growing as time passes by:

5. r r2 A = p

6. r A r 2 = p

7. r r 2 A = p

8. r r 2 A = p

9. A r 2 = p Both the radius r and the area Aare changing with time… … so the radius r and the area Aare functions of time… r

10. r2(t) A(t) = p Both the radius r and the area Aare changing with time… … so the radius r and the area Aare functions of time… r(t)

11. r2(t) A(t) = p Typical related rates question: • How are dA/dt and dr/dt related? • When the radius is 5 cm, increasing at a rate of 2 cm/sec, at what rate is area changing? r(t)

12. How are dA/dt and dr/dt related? If A and r are both functions of time, then

13. Without the t’s :

14. When the radius is 5 cm, increasing at a rate of 2 cm/sec, at what rate is area changing?

15. When the radius is 5 cm, increasing at a rate of 2 cm/sec, at what rate is area changing?

16. When the radius is 5 cm, increasing at a rate of 2 cm/sec, at what rate is area changing?

17. Water is being released from a spherical water balloon at a rate of 1000 cm3/minute. When the balloon’s radius is 10 cm, how fast is the balloon’s radius changing?

18. If V and r are both functions of time, then

19. Water is being released from a spherical water balloon at a rate of 1000 cm3/minute. When the balloon’s radius is 10 cm, how fast is the balloon’s radius changing?

20. Water is being released from a spherical water balloon at a rate of 1000 cm3/minute. When the balloon’s radius is 10 cm, how fast is the balloon’s radius changing?

21. Water is being released from a spherical water balloon at a rate of 1000 cm3/minute. When the balloon’s radius is 10 cm, how fast is the balloon’s radius changing?

22. Water is being released from a spherical water balloon at a rate of 1000 cm3/minute. When the balloon’s radius is 10 cm, how fast is the balloon’s radius changing?

23. Water is being released from a spherical water balloon at a rate of 1000 cm3/minute. When the balloon’s radius is 10 cm, how fast is the balloon’s radius changing?

24. Water is being released from a spherical water balloon at a rate of 1000 cm3/minute. When the balloon’s radius is 10 cm, how fast is the balloon’s radius changing? The ‘-’ sign tells us that the radius is decreasing.

25. One end of a 5 m ladder is sliding down a wall at a rate of 1 m/s. At what rate is the angle of the ladder with the floor changing when the other end is 3 m from the wall? 1 m/s 5 m h q

26. One end of a 5 m ladder is sliding down a wall at a rate of 1 m/s. At what rate is the angle of the ladder with the floor changing when the other end is 3 m from the wall? 1 m/s 5 m h q 3 m

27. We assume RADIAN measure If h and q are both functions of time, then

28. One end of a 5 m ladder is sliding down a wall at a rate of 1 m/s. At what rate is the angle of the ladder with the floor changing when the other end is 3 m from the wall?

29. 1 m/s 5 m h q 3 m One end of a 5 m ladder is sliding down a wall at a rate of 1 m/s. At what rate is the angle of the ladder with the floor changing when the other end is 3 m from the wall?

30. One end of a 5 m ladder is sliding down a wall at a rate of 1 m/s. At what rate is the angle of the ladder with the floor changing when the other end is 3 m from the wall?

31. One end of a 5 m ladder is sliding down a wall at a rate of 1 m/s. At what rate is the angle of the ladder with the floor changing when the other end is 3 m from the wall?

32. More Related Rates Examples

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

34. 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.

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

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

37. 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

38. 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 p