An oil refinery is located on the north bank of a straight river that is 1 km wide.

Question

A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 5 km east of the refinery. The cost of laying pipe is $300,000/km over land to a point P on the north bank and $600,000/km under the river to the tanks. To minimize the cost of the pipeline, how far (in km) from the refinery should P be located?

Answer 1

Let y = (5-x) be the distance from the refinery to point P. Let x = the distance from P to the Storage Tank.

Let c = 3000000 be the overland cost/km and 2c=6000000 be the under water cost/km.

Cost = C(x) = z(5-x) + 2z√(x²+1) where √(x²+1) is the distance under water

C'(x) = -cx +2cx/√(x²+1)

Setting C'(x) to zero and solving for x

2x = √(x²+1) The c value cancels out.

4x²=x²+1

3x²=1

x = √3/3

y = 5-√3/3 ≈ 4.23 Km, the distance from the refinery to point P.