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thedude23
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Now I hate optimization problems and I cannot figure this one out at all.
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?
b)
Just after construction has begun on the highway, the cost per km over marshland is reestimated
at $6 million per km. Find the point on the marsh/dryland boundary that would
minimize the total cost of the highway with the new cost estimate. How much would be saved
over continuing with the planned route?
My try:
Find the two triangles:
So 8 miles east and 4 miles south for the marsh.
8-x east and 6 miles south or the dryland.
So:
C(x)= (√(8^2-4^2))(5000000) + 2000000(√((8-x)^2-(6)^2))
Im pretty sure I am doing this wrong.
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?
b)
Just after construction has begun on the highway, the cost per km over marshland is reestimated
at $6 million per km. Find the point on the marsh/dryland boundary that would
minimize the total cost of the highway with the new cost estimate. How much would be saved
over continuing with the planned route?
My try:
Find the two triangles:
So 8 miles east and 4 miles south for the marsh.
8-x east and 6 miles south or the dryland.
So:
C(x)= (√(8^2-4^2))(5000000) + 2000000(√((8-x)^2-(6)^2))
Im pretty sure I am doing this wrong.