Related Rates

Greg Kelly, Hanford High School, Richland, Washington

Related Rates Problems

• All variables are differentiated with respect to TIME

• How quickly?• At what rate?• How fast is the _____________ changing?• Rate of change of ____________?• Write a general equation relating the variables,

then differentiate with respect to time.

Related Rates Problems

A ladder slides down the wall of a building as shown

y

x

How fast the height is changingdy

dt

How fast the distance from the base of the wall is changing

dx

dt

Suppose you were told that the bottom of a 13 foot tall ladder was 5 meters from the wall and moving away from the wall at 8 mps.

How fast is the top of the ladder moving down?

The ladder, the ground, and the wall make up a triangle

x

y13

The rate that the distance from the wall is changing

dxdt

8

The rate that the top of the ladder is falling

dydt

This is what we need to find!

x

y 13

What is the relationship between the three sides of the triangle?

2 2 169x y

2 2 0dx dyx ydt dt

But when x=5, y=12, and 8dx

dt

2 5 8 2 12 0dy

dt

3

10

dy

dt fps

WHY?

Suppose that the radius of a sphere is changing at an instantaneous rate of 0.1 cm/sec.(Possible if the sphere is a soap bubble or a balloon.)

310cm

dV

The change in radius per time

The change in volume per time

dV

dt

dr

dt

0.1 / seccm

Now we need a formula that relates the volume and the radius of a sphere

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

The formula for the volume of a sphere is

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

L3

sec

dV

dt

3cm3000

sec

Finddh

dt2V r h

2dV dhr

dt dt (r is a constant.)

32cm

3000sec

dhrdt

2

3000 cm

sec

dh

dt r

(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.

Hot Air Balloon Problem:

Given:4

rad

0.14min

d

dt

How fast is the balloon rising?

Finddh

dt

tan500

h

2 1sec

500

d dh

dt dt

2

1sec 0.14

4 500

dh

dt

h

500ft

Hot Air Balloon Problem:

Given:4

rad

0.14min

d

dt

How fast is the balloon rising?

Finddh

dt

tan500

h

2 1sec

500

d dh

dt dt

2

1sec 0.14

4 500

dh

dt

h

500ft

2

2 0.14 500dh

dt

1

12

4

sec 24

ft140

min

dh

dt

4x

3y

B

A

5z

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?

r t d 1

40 410

130 3

10

2 2 23 4 z 29 16 z

225 z5 z

4x

3y

30dy

dt

40dx

dt

B

A

5z

Truck Problem:

How fast is the distance between the trucks changing 6 minutes later?

r t d 1

40 410

130 3

10

2 2 23 4 z 29 16 z

225 z5 z

2 2 2x y z

2 2 2dx dy dzx y zdt dt dt

4 40 3 30 5dz

dt

250 5dz

dt

50dz

dt

miles50

hour

Truck A travels east at 40 mi/hr.Truck B travels north at 30 mi/hr.