The city wants to build a new stretch of highway to link an existing bridge with

Question

The city wants to build a new stretch of highway to link an existing bridge with an interchange, located 8 km to the south of the bridge. There is 5-km-wide stretch of marshland adjacent to the bridge that must be crossed (see the picture below). Given that the highway costs S19 million per kilometer to build over the marsh and only S7 million per kilometer to build over dry land, how far to the east of the bridge should the highway be when it crosses out the marsh (assume that the city wants to minimize the cost)?

Explanation / Answer

As in fig let its x km east when crossing Marshland.

Now Length in Marsh

L1 = (52 + x2 )1/2

Length In dry Land = L2 = (32 + (8-x)2)1/2 = ( 9 + 64 + x2 -16x)1/2

Total Cost ( in millions)

C = L1*19 + L2*7

=> C = (25 + x2)1/2 *19 + (73 + x2 -16 x)1/2*7

Now here x varies from 0 to 8

So

dC/dx = 19*(1/2)*(25+x2)-1/2 *2x + 7*(1/2)*(73+x2-16x)-1/2*(2x-16)

Points of Significance will be at

dC/dx = 0

Solving

We get x =1.76

So Crossing Point should be 1.76 km east

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