1

Related Rates Waner pg 329

Look at how the rate of change of one quantity is related to the rate of change of another quantity.

Ex. The radius of a circle is increasing at a rate of 10cm/sec. How fast is the area increasing at the instant when the radius has reached 5 cm?

Note: The rate of change of the area is related to the rate at which the radius is changing.

Lecture 15

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Solving Related RatesWaner pg 330

A. The problem

1. List the related, changing quantities.

2. Restate the problems in terms of rates of change. Rewrite the problem using mathematical notation for the changing quantities and their derivatives.

3

Rewriting inMathematical Notation

The population P is currently 10,000 and growing at a rate of 1000 per year.

rpeople/yea1000dt

dP

people000,10P

4

Rewriting inMathematical Notation

There are presently 400 cases of Bird flu, and the number is growing by 30 new cases every month.

flu bird of cases400B

hcases/mont30dt

dB

5

Rewriting inMathematical Notation

The price of shoes is rising $5 per year. How fast is the demand changing?

$/year5dt

dp

$/year?dt

dD

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Solving Related RatesWaner pg 330

A. The problem

1. List the related, changing quantities.

2. Restate the problems in terms of rates of change. Rewrite the problem using mathematical notation for the changing quantities and their derivatives.

B. The relationship

1. Draw a diagram, if appropriate.

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2. Find an equation relating the changing quantities.

3. Take the derivative with respect to time of the equation(s) to get the derived equation(s), relating the rates of change of the quantities.

C. The solution

1. Substitute into the derived equation(s)

2. Solve for the derivative required.

Solving Related Rates (cont)

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Ex. Two cars leave from an intersection at the same time. One car travels north at 35 mi./hr., the other travels east at 60 mi./hr. How fast is the distance between them changing after 2 hours?

Distance = z

x

y

2 2 2x y z

mi/hr60dt

dx

mi/hr35dt

dy

miles120xmiles70y

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Distance = z

x

y

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

From original relationship:

69.5 mi./hr.dz

dt

How fast is the distance between them changing after 2 hours?mi/hr60

dt

dx

mi/hr35dt

dy

miles120x

miles70y22 yxz

dt

dz)92.138(2)35)(70(2)60)(120(2

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Two cars leave from an intersection at the same time. One car travels north at 35 mi./hr., the other travels west at 60 mi./hr. How fast is the distance between them changing after 2 hours?

Answer:

The instantaneous change in distance between the two cars with respect to time after 2 hours is 69.5 miles per hour.

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Related RatesWaner pg 334, #19

Demand Assume that the demand equation for tuna in a small costal town is

(pounds) tunaof demandmonthly

($) poundper price :where

000,505.1

q

p

pq

The town’s fishery finds that the monthly demand for tuna is currently 900 pounds and increasing at a rate of 100 pounds per month each month. How fast is the price changing?

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Related RatesWaner pg 334, #9

Sunspots The area of a circular sunspot is growing at a rate of 1200 km2/sec.

a. How fast is the radius growing at the instant when it equals 10,000 km?

b. How fast is the radius growing at the instant when the sunspot has a area of 640,000 km2?

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Related RatesWaner pg 334, #11

Sliding Ladders The base of a 50 foot ladder is being pulled away from a wall at a rate of 10 feet per second. How fast is the top of the ladder sliding down the wall at the instant when the base of the ladder is 30 feet from the wall?

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Related RatesWaner pg. 336, #35

Cylinders The volume of paint in a right cylindrical can is given by V = 4t2 – t where t is time in seconds and V is the volume in cm3. How fast is the level rising when the height h is 2 cm? The can has a total height of 4 cm and a radius r of 2 cm. (Volume of a cylinder is given by V = πr2h.)