Related Rates

1. Related Rates

2. Expanding Balloon Air is being pumped into a spherical balloon at the rate of 100 cm3/sec (so the volume of the balloon is increasing at the rate of 100 cm3/sec). What is the rate of change of the radius of the balloon?

3. Expanding Balloon Air is being pumped into a spherical balloon at the rate of 100 cm3/sec (so the volume of the balloon is increasing at the rate of 100 cm3/sec). What is the rate of change of the radius of the balloon?

4. Expanding Balloon Air is being pumped into a spherical balloon at the rate of 100 cm3/sec (so the volume of the balloon is increasing at the rate of 100 cm3/sec). What is the rate of change of the radius of the balloon?

5. What We Know: The rate of change of the volume of the balloon with respect to time is 100 cm3/sec The relationship between the volume of a sphere and its radius

6. What We Want to Know: The rate of increase of the radius of the balloon with respect to time

7. How Do We Solve This? Using implicit differentiation differentiate both sides with respect to time (t)

8. Solution: Notice that the rate of change of the radius with respect to time depends on the actual value that the radius has at that time. What is the rate of change of the radius when r=10 cm ?

9. Sliding Ladder A 13 foot long ladder is leaning against a vertical wall and begins to slide down the wall at a rate of .5 ft/sec. How fast is the foot of the ladder moving away from the wall? wall -.5 ft/sec ladder 13 ft ground ?

10. wall (a) -.5 ft/sec ladder 13 ft (c) ground ? (b) What We Know The length of the ladder - 13 ft The relationship between the length of the ladder and the length of the sides c2=a2+b2 169=a2+b2 The rate at which the top of the ladder is sliding down the wall

11. wall (a) -.5 ft/sec ladder 13 ft (c) ground ? (b) What We Want to Know The rate at which the foot of the ladder is moving away from the wall (which equals the rate at which b is changing with respect to time).

12. Solution We solve this the same way we solved the earlier problem - by using implicit differentiation with respect to time (t) At what rate is the foot of the ladder moving away from the wall when it is 5 feet from the wall?

13. wall (a) 8 ft -.5 ft/sec ladder 13 ft (c) ground ? (b) At what rate is the foot of the ladder moving away from the wall when the top of the ladder is 8 feet from the ground?

14. 4 mi b ? c Flying Plane A plane flying at a height of 4 miles above the ground and a speed of 500 mph passes directly over your house. At what rate is the distance between your house and the plane changing? a 500mph

15. a 500mph 4 mi b ? c What We Know b= 4 miles The relationship between a, b and c c2=a2+b2 c2=a2+16 The rate at which the plane is moving away from the house

16. a 500mph 4 mi b ? c What We Want to Know The rate at which c is changing with respect to time.

17. Solution As before, differentiate implicitly with respect to time. At what rate is the plane moving away from the house when it is 3 miles past the house?

18. 8 cm 16 cm Conical Cup A paper cup in the shape of an inverted right circular cone has a depth of 16 centimeters and a radius of 8 centimeters. Water is being poured into the cup at a rate of 4 cm3 per second. At what rate is the depth of the water changing?

19. What We Know The volume of the cup At any given water depth the volume of water is where h is the depth of the water and r is the radius of the surface area of the water. The relationship between h and r is Rewriting the formula for the volume of the water r h

20. More What We Know The rate at which the volume of water is increasing is 4 cm3/min.

21. What We Want to Know The rate at which the depth (h) of the water is increasing

22. Solution Differentiate implicitly with respect to time and solve for What is the rate of change of the water depth when the water is 3 cm deep?

23. Growing Bubble A spherical bubble is growing so that its surface area increases by 2 in2/min. At what rate is the diameter of the bubble increasing?

24. What We Know The surface area of the bubble is given by where x is the diameter of the bubble The rate of change of the surface area with respect to time

25. What We Want to Know The rate at which the diameter is changing with respect to time

26. Solution Differentiate both sides with respect to time (t) and solve for What is the rate of change of the diameter when the diameter equals 3 in?

27. The Dukes of Hazzard Beau and Luke Duke are speeding through the back roads of Hazzard County trying to get away from Boss Hogg. They realize they will have to jump the “General Lee” over Hazzard Crick. Fortunately there is a ramp at the edge of the crick for just this purpose. The length of the ramp is 100 feet and the angle of elevation at the base of the ramp is 200. If the speed of the “General Lee” is 60 mph on the ramp how fast is the car rising vertically at the instant when it leaves the ramp? What we want to know is what is the rate of change in b when the car leaves the ramp

28. What We Know The relationship between b, c and ө The rate at which the car is moving up the ramp

29. What We Want to Know We want to know the rate at which the car is rising above the ground when it leaves the end of the ramp. ?

30. Solution Differentiate the equation implicitly with respect to t. Remember that ө is a constant and so sin(ө) is also a constant, and not changing with time. The car is rising vertically at a rate of 30.396 ft/sec at the moment when it leaves the end of the ramp.

31. Summary When solving these problems 1. Try to draw a picture. 2. Write down what you know - either given in the problem or general knowledge. 3. Write down what you want to find out. 4. Find the relationship between what you know and what you want to find out. 5. Solve!