An oil refinery is located on the north bank of a straight river that is 2 km wide. (cont.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. minimize the cost how far from the refinery should P be located?

Answer 1

Draw and label a diagram!

R is the location of the refinery

O is point opposite the refinery

P is the desired point

S is the location of the storage tank

PS = x, the land pipe

OP = 7 - x

2² + (7 - x)² = RP²

RP² is the underwater pipe

Total pipe = RP² + x

Pipe Cost = 8RP² + 4x (in $100,000's)

Pipe Cost = 8(2² + (7 - x)²) + 4x

Pipe Cost = 8x² - 116x + 424

Minimum occurs at x = -b / 2a, or 7.25