BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071
BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

Solutions

Chapter 1.6, Problem 83E
To determine

To evaluate: The length of abandoned road used to make the new road and whether the cost to make new road directly to connect the town is less than $6.8 million.

Expert Solution

Answer to Problem 83E

The length of abandoned road used to make the new road is 16 mi and the cost to make new road directly to connect the town is not less than $6.8 million.

Explanation of Solution

Given:

The town Foxton lies 10 mi north of and abandoned east-west road. The distance between the point on the abandoned road closest to Foxton is 40 mi from Grimley.

The cost for restore the old road is $100,000 and cost to make new road is $200,000.

The total money spend to make the road is $6.8 million.

The total money spend to make the road is $6.8 million.

Cost for new road is $200,000.

Calculation:

Let the length of abandoned road used to make the new road be x and the remaining part of road be (40x) .

Length of new road is,

(hypotenuse)2=(base)2+(height)2(hypotenuse)2=(40x)2+(10)2(hypotenuse)=(40x)2+(10)2

Tabulate the given information into the language of algebra.

In words In algebra
Length of abandoned road use to make new road x
Remaining part of abandoned road 40x
Length of new road (10)2+(40x)2
Old cost road 100x
New cost road 200[(10)2+(40x)2]

Model the equation for the above information.

Oldroadcost+Newroadcost=6.8million100(x)+200[(10)2+(40x)2]=6800

Simplify the above equation for x.

100x+200(100+1600+x280x)=6800x+2(x280x+1700)=68x280x+1700=68x2

Take square of both sides of above equation.

x280x+1700=(68x2)2x280x+1700=4624+x2136x44x2320x+6800=4624+x2136x3x2184x+2176=0

Factor the above equation.

(3x136)(x16)=0x=1363,x=16

Thus, length of abandoned road used to make the new road is 16 mi.

The length of new road connecting two towns directly is,

(hypotenuse)2=(base)2+(height)2(hypotenuse)2=(40)2+(10)2(hypotenuse)2=1700hypotenuse=41.231

The length of new road connecting two towns directly is 41.231 mi.

The cost for new road is,

Cost=200×41.231=8246thousand=8.246million

Thus, the cost to make new road directly to connect the town is not less than $6.8 million.

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