Minimum Cost An offshore oil well is kilometers off the coast. The refinery is kilometers down the coast. Laying pipe in the ocean is twice as expensive as laying it on land. What path should the pipe follow in order to minimize the cost?

To minimize cost, the pipe should follow a rectangular path.

Explanation:

Let us denote the square bottom of the dug pit as

P = S²

and its cost will be P = 2S² = 4S to dig

Let us denote the height of the pit as H,

Therefore, Total area of the pit will be:

4SH

and its cost is 2SH

The volume of the pit is

v = S²H = 4² * 2 = 128

Total cost therefore is =

C = 2S² + 2SH

This therefore translates to

128 = s²h

Making H the subject of the formula,

we have, h = 128 / S²

Cs therefore = 2S² + 2S. 128 / S²

Cs = 2S² + 256 / S

Cs = 4S - 256 / S²

Cs = 4S³ - 256 / S

4S³ - 256 / S = 0

4S³ - 256 - 0 = S

Solving further, S = 4

Where,

H = 128/64

= 2

Therefor, S = 4 and H = 2

So our pipe will follow a rectangular path to properly minimize cost.