SOLUTION: The town of Foxton lies 10 mi north of an abandoned east-west road that runs through Grimley. The point on the abandoned road closest to Foxton is 40 east of Grimley. County offi

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Question 214882: The town of Foxton lies 10 mi north of an abandoned east-west road that runs through Grimley. The point on the abandoned road closest to Foxton is 40 east of Grimley.
County officials are about to build a new road connecting the two towns. They have determined that restoring the old road would cost $100,000 per mile, whereas building a new road would cost $200,000 per mile. How much of the abandoned road should be used if the officials intend to spend $6.8 million? Would it cost less than this amount to build a new road connecting the towns directly?
Answer by ankor@dixie-net.com(22740) (Show Source):
You can put this solution on YOUR website!
The town of Foxton lies 10 mi north of an abandoned east-west road that runs through Grimley.
The point on the abandoned road closest to Foxton is 40 east of Grimley.
County officials are about to build a new road connecting the two towns.
They have determined that restoring the old road would cost $100,000 per mile,
whereas building a new road would cost $200,000 per mile.
How much of the abandoned road should be used if the officials intend to spend $6.8 million?
;
Drawing a diagram of this will make it easy to understand.
:
Let x = the unused portion of the old road from a point south of Foxton
then
(40-x) = the portion of the old road to be used.
That's the point where it begins the new road diagonally to Foxton
:
This diagonal is the hypotenuse of a right triangle formed by sides 10 and x
:
The equation using 1000's of dollars
:
old road cost + new road cost = 6.8 million
100(40-x) + 200() = 6800
:
4000 - 100x + 200() = 6800
:
Get the square root by itself on the left
200() = 100x + 6800 - 4000
:
200() = 100x + 2800
square both sides
40000(x^2 + 100) = (100x + 2800)^2
:
40000x^2 + 4000000 = 10000x^2 + 560000x + 7840000
:
0 = 10000x^2 - 40000x^2 + 560000x + 7840000 - 4000000
:
-30000x^2 + 560000x + 3840000 = 0
Simplify divide by -10000 (change the signs)
3x^2 - 56x - 384 = 0
Factor this to:
(3x + 16)(x - 24) =
Positive solution
x = 24 mi of unused old road
then
40 - 24 = 16 mi of the old road used at 100k/mi
:
New road dist, find the hypotenuse:
NR =
NR = 26 mi of new road at 200k/mi
;
:
Check the cost, new road + old road:
Cost = 200(26) + 100(16)
Cost = 5200 + 1600
Cost = 6800 thousand which is 6.8 million
:
:
Would it cost less than this amount to build a new road connecting the towns
:
No
cost = 200*
Cost = 200 * 41.231 mi
Cost = 8246 thousand which is 8.246 million
;
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Actually you could find the minimum cost by finding the vertex of the equation, but they are determined to spend 6.8 million, how like the government!!