Related Rates5.6

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?

34

3V r

24dV r dr

24 10cm 0.1cmdV

340 cmdV

The volume would change by approximately .340 cm

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.)

34

3V r

24dV dr

rdt dt

2 cm4 10cm 0.1

sec

dV

dt

3cm

40sec

dV

dt

The sphere is growing at a rate of .340 cm / sec

Note: This is an exact answer, not an approximation like we got with the differential problems.

Water is draining from a cylindrical tank at 3 liters/second. How fast is the surface dropping?

L3

sec

dV

dt

3cm3000

sec

(r is a constant.)

(We need a formula to relate V and h. )

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 .

Hot Air Balloon Problem:

Given:4

rad

0.14min

d

dt

How fast is the balloon rising?

h

500ft

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?

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=(πr2h)/3 of the cone. Find an equation that relates dV/dt, dr/dt, and dh/dt.

A Highway CaseA 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?

Example A Highway Chase (cont’d)

Filling a Conical Tank

Water runs into a conical tank at the rate of 9 ft3/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.

The sides of a right triangle with legs and and hypotenuse

increase in such a way that / 1 and / 3 / . At the instant

when 4 and 3, what is / ?

(A) 1/3

(B) 1

(C) 2

(D) 5

(E) 5

x y z

dz dt dx dt dy dt

x y dx dt

An observer 70 meters south of a railroad crossing watches an eastbound

train traveling at 60 meters per second. At how many meters per second is the

train moving away from the observer 4 seconds after it passes through the

intersection?

(A) 57.60

(B) 57.88

(C) 59.20

(D) 60.00

(E) 67.40