Unit 3 - Objective 3 - Related Rates

Example 1

The derivative also represents rate of change. Do the following for related rate problems:

• Read slowly.
• Make a picture if possible.
• Translate into math sentences.
• List what is given and what you are asked to find.

Try it!

The voltage of a certain thermocouple as a function of temperature is given by

E = 2.800T + 0.006T²

If the temperature is increasing at the rate of 0.5° C/min, how fast is the voltage increasing when T = 100°C?

Given: E = 2.8T + 0.006T² Find: when T=100° C Solution: E = 2.8T + 0.006T² use implicit differentiation = 2.8 + 2 (0.006) T Now substitute what is given = 2.8 (0.5) + 2 (0.006) (100) (0.5) = 1.4 + 0.6 = 2

Example 2

To solve related rate problems (after translating "given" and "to find"):

1. Write an equation which relates the variables whose derivatives are given and those whose derivatives are asked for.
2. Differentiate each term of that equation implicitly with respect to t (time).
3. Substitute in given values.
4. Solve for expression asked for.

Try it!

Problem 1:

A spherical balloon is being blown up at a constant rate of 2 ft³/min. Find the rate at which the radius is increasing when the radius is 3 feet.

Rate means derivative, so translation becomes:

Given: = 2 ft³/min (ft³ means volume) Find: = ? when r = 3 ft

First we need an equation relating v and r (see inside front cover of text):

1. Equation for volume of sphere:
2. Differentiate equation (with respect to t)
3. Substitute in given
4. Solve for dr/dt

Problem 2:

A 15 foot ladder is sliding down the side of a vertical building. If the top of the ladder is moving at the rate of .25 ft/min, how fast is the foot of the ladder moving when it is 9 feet from the building?

Given: dx/dt = -0.25 ft/min (negative because the ladder is moving down the wall) Find: dy/dt when y = 9 x² + y² = 15² differentiate (with respect to t) 2x + 2y = 0

You need to find the value for x before you substitute into this equation:

x² + 9² = 15² x² + 81 = 225 x² = 144 x = 12

(only positive values apply)

Now since: