Easy Optimization problem on math??

Posted on 2015-11-14 06:18:00 in Math by BONNY

1) A city wants to build a new section of highway to link an
existing bridge with an existing highway interchange, which
lies 8 miles to the east and 10 miles to the south of the bridge.
The first 4 miles south of the bridge is marshland. Assume
that the highway costs $5 million per mile over marsh and
$2 million per mile over dry land. The highway will be built
in a straight line from the bridge to the edge of the marsh,
then in a straight line to the existing interchange.

a) At what point should the highway emerge from the marsh in order to
minimize the total cost of the new highway? How much is
saved over building the new highway in a straight line from the
bridge to the interchange? (Hint: Use similar triangles to find
the point on the boundary corresponding to a straight path and
evaluate your cost function at that point.)

2) After construction has begun on the highway in exercise 19, the
cost per mile over marshland is reestimated at $6 million. Find
the point on the marsh/dry land boundary that would minimize
the total cost of the highway with the new cost function. If the
construction is too far along to change paths, how much extra
cost is there in using the path from exercise 19?

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