mth095 intermediate algebra chapter 7 – rational expressions sections 7.6 – applications and...

MTH095MTH095Intermediate AlgebraIntermediate Algebra

Chapter 7 – Rational Expressions

Sections 7.6 – Applications and Variations Motion (rate – time – distance) Shared Work Variation (direct, inverse, & joint)

Copyright © 2010 by Ron Wallace, all rights reserved.

MotionMotion

Example …The current in the Lazy River moves at 4

mph. Monica’s dinghy motors 6 miles upstream in the same time it takes to motor 12 miles downstream. What would be the speed of her dinghy in still water?

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MotionMotion



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Motion d r t

MotionMotion



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Motion d r tUpstream

MotionMotion



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Motion d r tUpstream 6

MotionMotion



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Motion d r tUpstream 6 r – 4

MotionMotion



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Motion d r tUpstream 6 r – 4 t

MotionMotion



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Motion d r tUpstream 6 r – 4 tDownstream

MotionMotion



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12

MotionMotion



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12 r + 4

MotionMotion



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12 r + 4 t

MotionMotion



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Motion d r tUpstream 6 r – 4 t = 6/(r –

4)Downstream

12 r + 4 t = 12/(r + 4)

MotionMotion



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6 12

4 4r r

Time Up = Time Down

12 mphr

MotionMotion

In general … begin by filling in the table ...

Use a formula to eliminate a variable.Set equal expressions equal to each

other.Solve & Check

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Motion d r tMotion 1 ? ? ?

Motion 2 ? ? ?

Motion – Example Motion – Example

A doctor drove 200 miles to attend a national convention. Because of poor weather, her average speed on the return trip was 10 mph less than her average speed going to the conventions. If the return trip took 1 hour longer, how fast did she drive in each direction?

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Motion d r tGoing there

Coming home

Motion – Example Motion – Example

A doctor drove 200 miles to attend a national convention. Because of poor weather, her average speed on the return trip was 10 mph less than her average speed going to the conventions. If the return trip took 1 hour longer, how fast did she drive in each direction?

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Motion d r tGoing there 200 r t1= 200/r

Coming home

200 r – 10 t2= 200/(r – 10)

Shared WorkShared WorkExample …

Tom & Sue work for the city parks department where they mow the lawn in the city park. Tom, working by himself, can mow the lawn in 5 hours. Sue, working by herself, can mow the lawn in 4 hours. How long will it take to mow the lawn if they work together?

Estimates? 9 hours? 4.5 hours? 2.25 hours? Other guesses?

Tom Tom (working alone)(working alone) takes 5 takes 5 HoursHours


Hour#1


Hour#1

Hour#2


Hour#1

Hour#2

Hour#3


Hour#1

Hour#2

Hour#3

Hour#4


Hour#1

Hour#2

Hour#3

Hour#4

Hour#5

Sue Sue (working alone)(working alone) takes 4 takes 4 HoursHours


Hour#1


Hour#1

Hour#2


Hour#1

Hour#2

Hour#3


Hour#1

Hour#2

Hour#3

Hour#4

Working TogetherWorking TogetherTom takes 5 hours.• 1/5 of the job each hour.

Sue takes 4 hours.• 1/4 of the job each hour.

Working TogetherWorking Together

Hour#1

Hour#1

Tom takes 5 hours.• 1/5 of the job each hour.



Hour#1

Hour#1

Hour#2

Hour#2



Working TogetherWorking TogetherTom takes 5 hours.• 1/5 of the job each hour.


Hour#1

Hour#1

Hour#2

Hour#2

Hour #3 ?


Let x = # hours to complete the job together.

Portion of work completed by Tom

Portion of work completed by Sue

Adding these gives …



1

5x

1

4x

1 11

5 4x x 2

9

20 hr 2 hr

9x

Shared Work – Summary Shared Work – Summary

Time Individual Time Individual Time Individual

Working #1 Rate Working #2 Rate Working #3 Rate

1Individual 1

Time to completeRate

the task alone

Proportions – Equality of Proportions – Equality of RatiosRatios

Ratio: A quotient of related quantities.

Proportion: Two equivalent rations that relate the same quantities.

One of the quantities will be unknown.

Does it matter which quantity is on top?

A C

B D


Example …An automobile gets 23 miles per gallon of

gas (mpg). How much gas does it take to travel 200 miles?

The Ratio …

A C

B D

miles

gallons


Example …An automobile gets 23 miles per gallon of

gas (mpg). How much gas does it take to travel 200 miles?

The Ratio …

A C

B D

23

1

miles

gallons

200

miles

x gallons

8.7 x gallons

Two Common ProportionsTwo Common Proportions

Similar Triangles◦Corresponding Angles are Equal◦Ratios of Corresponding Sides are

Equal

Scale Drawings

◦Maps

◦Blueprints

inches

miles

inches

feet

VariationVariation

Direct Variation Two quantities whose ratio is a

constant.

Inverse Variation Two quantities whose product is a

constant.

Others – Combinations of the Above

◦e.g. Joint Variation Three quantities where the ratio of

one of the quantities to the product of the other two quantities is a constant.

Direct VariationDirect Variation

Two quantities whose ratio is a constant.

“ y varies directly as x ” aka: “y is [directly] proportional to x”

k is called the “constant of proportionality”

yk

x y kx

In an application, data is given to determine k and then values of x are used to determine values of y.

Direct VariationDirect Variation

Example ...The weight hanging from a spring is

directly proportional to the distance the spring is stretched (Hooke’s Law). If a 6 pound weight stretches a particular spring 5 inches, and a fish hanging from the same spring stretches the spring 9 inches, how much does the fish weigh?

W kS 6 5k 1.2k

1.2W S 1.2(9)W 10.8 lbsW

Inverse VariationInverse Variation

Two quantities whose product is a constant.

“ y varies inversely as x ” aka: “y is inversely proportional to x”

k is called the “constant of proportionality”

xy k ky

x

In an application, data is given to determine k and then values of x are used to determine values of y.

Inverse VariationInverse Variation

Example ...The time it takes a to get sunburned

varies inversely with the UV rating on that day? If a UV rating of 4 causes person with fair skin to burn in 20 minutes, how long will it take for them to burn on a day with a UV rating of 7?k

TU

204

k 80k

80T

U

80

7T 11.4 minT

http://www.revolutionhealth.com/articles/uv-index/stu3205

mth095 intermediate algebra chapter 7 – rational expressions sections 7.6 – applications and...

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