An oil refinery is located on the north bank of a straight river that is 2 km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 3 km east of the refinery. The cost of laying pipe is $300,000 per km over land to a point P on the north bank and $600,000 per km under the river to the tanks. To minimize the cost of the pipeline, how far from the refinery should P be located? (Give your answer correct to two decimal places.)

Answer 1

Let x be the distance between P and a point on the north bank directly opposite to the tanks.

The length of the pipe under water,
L1(x) = √(2²+x²).
The length of the pipe on land,
L2(x) = 3-x
Cost of laying the pipes,
C(x) = 600000 L1(x) + 300000 L2(x)

Differentiate C(x) with respect to x and equate the derivative to zero.
Solve for x.
The distance from the refinery to P is (3-x).
I get about 1.8 km.