1. Related Rates 5.6

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 0.1 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. How fast is the surface dropping? ( r is a constant.) (We need a formula to relate V and h. )

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 – PLUG IN WHAT YOU KNOW AFTER DIFFERENTIATING.

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

7. 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?

8. Related Rates Assume that the radius r and height h of a cone are differentiable functions of t and let V be the volume V=(πr 2 h)/3 of the cone. Find an equation that relates dV/dt, dr/dt, and dh/dt.

9. A Highway Case A police cruiser, approaching a right-angled intersection from the north, is chasing a speeding car that has turned the corner and is now moving straight east. When the cruiser is 0.6 mi north of the intersection and the car is 0.8 mi to the east, the police determine with radar that the distance between them and the car is increasing at 20 mph. If the cruiser is moving at 60 mph at the same instant of measurement, what is the speed of the car?

10. Example A Highway Chase (cont’d)

11. Filling a Conical Tank Water runs into a conical tank at the rate of 9 ft 3 /m. The tank stands point down and has a height of 10 ft and a base radius of 5 ft. How fast is the water level rising when the water is 6 ft deep.