Construction Costs 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 40
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, whereas 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 $6.8 million? Would it cost less
than this amount to build a new road connecting the towns directly?
2 answers
No figure. Cannot copy and paste here.
the length of a direct road is √(40^2 + 10^2) = √1700 = 41.23, and its cost is thus about $.2M*41.23 = $8.246M, more than the budget allows
If x miles of the old road are used, the distance is
x + √((40-x)^2 + 10^2)
so to spend the budgeted amount, you want (suppressing all the extra 5 zeros)
x + 2√((40-x)^2 + 10^2) = 68
x = 16 mi
If x miles of the old road are used, the distance is
x + √((40-x)^2 + 10^2)
so to spend the budgeted amount, you want (suppressing all the extra 5 zeros)
x + 2√((40-x)^2 + 10^2) = 68
x = 16 mi