SOLUTION: The town of Foxton lies 10 mi north of an abandoned east-west road that runs through Grimley, as shown in the figure. The point on the abandoned road closest to Foxton is 70 mi fro

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Question 1137815: The town of Foxton lies 10 mi north of an abandoned east-west road that runs through Grimley, as shown in the figure. The point on the abandoned road closest to Foxton is 70 mi from 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, while building a new road would cost $200,000 per mile.
How much of the abandoned road should be used (as indicated in the figure) if the officials intend to spend exactly $9.8 million?
Would it cost less than this amount to build a new road connecting the towns directly?
Answer by ankor@dixie-net.com(22740) About Me  (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, as shown in the figure.
The point on the abandoned road closest to Foxton is 70 mi from 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, while building a new road would cost $200,000 per mile.
How much of the abandoned road should be used (as indicated in the figure) if the officials intend to spend exactly $9.8 million?
:
Let x = dist from closest point on the old road to Foxton to the point where the new road takes off for Foxton
then
(70-x) = the distance used on the old road
and the dist used on the new road will be the hypotenuse of 10 and (70-x)
the equation using million$, .1 for old road .2 for new road per mi
:
old road cost + new road cost = 9.8
.1%2870-x%29+%2B+.2%28sqrt%2810%5E2+%2B+x%5E2%29%29+=+9.8
graph the equation
y = .1%2870-x%29+%2B+.2%28sqrt%2810%5E2+%2B+x%5E2%29%29+-+9.8

x = 24
Therefore 70-24 = 46 mi of the abandoned road should be used
Check: .1(46) + .2(sqrt%2810%5E2%2B24%5E2%29) = 9.8
:
Would it cost less than this amount to build a new road connecting the towns directly?
No, .2%28sqrt%2810%5E2%2B70%5E2%29%29 > 14 million
:
Sorry about this, I got called away suddenly yesterday and inadvertently sent it unfinished. 'CK