Related Rates

Lesson 6.5

General vs. Specific

Note the contrast …General situation• properties true at every instant of time

Specific situation• properties true only at a particular instant of

time

We will consider a rock dropped into a pond … generating an expanding ripple

2

Expanding Ripple

At the point in time whenr = 8• radius is increasing

at 3 in/sec• That is we are given

We seek the rate that the area is changing at that specific time

• We want to know

3

r = 83dr

dt

dA

dt

Solution Strategy

1. Draw a figure label with variables do NOT assign exact values

unless they never change in the problem

2. Find formulas that relate the variables

4

Ar

2A r 3dr

dt

Solution Strategy

3. Differentiate the equation with respect to time

4. Substitute in the given information

5

2dA dr

rdt dt

8

3

r

dr

dt

22 8 3 48 in / sec

Example

Given

Find when x = 3

Note: we must differentiate implicitly with respect to t

6

2 2 25 4dx

x ydt

dy

dt

2 2 0dx dyx ydt dt

Example

Now substitute in the things we know

• x = 3

Find other values we need• when x = 3,

32 + y2 = 25 and y = 4

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2 2 0dx dyx ydt dt

4dx

dt

Example

Result

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2 2 0dx dyx ydt dt

2 4 4 2 4 0

324

8

dy

dtdy

dt

Particle on a Parabola

Consider a particle movingon a parabola y2 = 4x at (1,-2)

Its horizontal velocity(rate of changeof x) is 3ft/sec

What is the vertical velocity,the rate of change of y?

9

•

3dx

dt

??dy

dt

Particle on a Parabola

Differentiate the original equation implicitly with respect to t

Substitute in the values known

Solve for dy/dt

10

Draining Water Tank

Radius = 20, Height = 40

The flow rate = 80 gallons/min

What is the rate of change of the radius when the height = 12?

11

21

3Volume r h

80dV

dt

??dr

dt

Draining Water Tank

At this point in timethe height is fixed

Differentiate implicitly with respect to t, Substitute in known valuesSolve for dr/dt

12

2112

3Volume r

12 12

3

dV drr

dt dt

Assignment

Lesson 6.5

Page 409

Exercises 1 – 27 odd

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