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Related Rates and Applications

Lesson 3.7

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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

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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

r = 8

3dr

dt

dA

dt

View Spreadsheet demonstration

View Spreadsheet demonstration

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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

Ar

2A r 3dr

dt

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Solution Strategy

3. Differentiate the equation with respect to time

4. Substitute in the given information

2dA dr

rdt dt

8

3

r

dr

dt

22 8 3 48 in / sec

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Example

• Given

• Find when x = 3

Note: we must differentiate implicitly with respect to t

2 2 25 4dx

x ydt

dy

dt

2 2 0dx dyx ydt dt

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Example

• Now substitute in the things we know

– x = 3

• Find other values we need– when x = 3,

32 + y2 = 25 and y = 4

2 2 0dx dyx ydt dt

4dx

dt

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Example

• Result

2 2 0dx dyx ydt dt

2 4 4 2 4 0

324

8

dy

dtdy

dt

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Guidelines for Related-Rate Problems

1. Identify given quantities, quantities to be determined

• Make a sketch, label quantities

2. Write equation involving variables

3. Using Chain Rule, implicitly differentiate both sides of equation with respect to t

4. After step 3, substitute known values, solve for required rate of change

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Electricity

• The combined electrical resistance R of R1 and R2 connected in parallel is given by

• R1 and R2 are increasing at rates of 1 and 1.5 ohms per second respectively.

• At what rate is R changing when R1 = 50 and R2 = 75?

1 2

1 1 1

R R R

R1R1

R2R2

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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?

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3Volume r h

80dV

dt

??dr

dt

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Draining Water Tank

• At this point in timethe height is fixed

• Differentiate implicitly with respect to t,

• Substitute in known values

• Solve for dr/dt

2112

3Volume r

12 12

3

dV drr

dt dt

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Assignment

• Lesson 3.7

• Page 187

• Exercises 1 – 7 odd, 13 – 27 odd