This problem is really confusing me..please help!?

Question

This is an optimization problem in calculus.

An offshore oil well is 2 kilometers off the coast. The refinery is 4 kilometers down the coast. If laying pipe in the ocean is twice as expensive as on land, what path should the pipe follow in order to minimize the cost?

What are the steps to arrive at this answer and what is the final answer? Any and all help would be greatly appreciated!

Answer 1

draw a sketch: a vertical line for the coast, a point to the right for the well (W), a point on the line closest to the well (A), a point on the line below that for the refinery (R), and a point in between A and R, B. draw WB. label AW with a 2, AB with an x, BR with 4-x. BW, the underwater route, is √(x²+4). assume generally part of the route is underwater, part maybe on land depending on the value of x. cost will be
C = 1(4-x) + 2√(x² + 4)
and it is minimum when C' = 0, given that
C' = -1 + 2/[-2√(x² + 4)](2x)
C' = -1 - 2x/√(x² + 4) = 0
1 = -2x/√(x² + 4)
√(x² + 4) = -2x
x² + 4 = 4x
3x² = 4
x² = 4/3
x = 2/√3 = 1.1547

so cheapest route runs underwater to a point 1.1547 km down the coast, then on land the rest of the way.

Answer 2

Think about it. basic arithmetic here. Its a trick question. two times the cost of two kilometers would be the same price as 4 kilometers if the path is being measured and payed for in units of kilometers/