An oil refinery is located on the north bank of a straight river that is 2 km wide. (continue in description)

Question

A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 7 km east of the refinery. The cost of laying pipe is $400,000/km over land to a point P on the north bank and $800,000/km under the river to the tanks. To minimize the cost of the pipeline, how far from the refinery should P be located? (Round your answer to two decimal places.)

Answer 1

Let x = distance over land from the refinery and y = distance through water.

y will be a diagonal going south to north forming a right triangle

Total Cost = 400000x + 800000y Or C = x + 2y (Twice expensive to go thru water)

y² = 2² + x²

Take the derivative of the cost:

C ' = 1 + 2y '

y = (4 + x²)^1/2

y ' = (1/2)2x/(4 + x²)^1/2

Setting the derivative of the cost to zero

1 + 2x/(4 + x²)^{1/2= 0}

(4 + x²)^1/2 = -2x

4 + x² = 4x²

3x² = 4 --> x = ± √(4/3) Take the positive one, so x = 2√3/3 ≅ 1.15 km

Set Point P to 1.15 km east of the refinery on the north side of the river