Calculus word problem

Question

An offshore oil well is 2 km off the coast. The oil refinery is 4 km 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 cost?

Answer 1

Draw a picture first.

Let x be the length of the land segment. Then, we have a right triangle where the hypotenuse is the length of the ocean segment---i.e. the distance from the well to the end of the land segment. One side is the direct line from the well to land (2km), and the third side is 4-x. Using the pythagorean theorem, we can write the equation for the ocean segment:

√(2² + (4-x)²)

Now, write the equation for the cost: Assume that the land cost is 1, and therefor the ocean cost is 2

Cost, C(x) = x + 2 * √(2² + (4-x)²)

To find the minimum C, we take the 1st derivative and set it equal to 0:

Rewrite C(x) = x + 2 * (4 + (4-x)²)^1/2

C'(x) = 1 + 2 * (4 + (4-x)²)^-1/2(2(4-x)*(-1))

Set this = to 0, rearrange and simplify and solve for x. One of the solutions will be the optimum value for x.

EDIT: I get 2.845 and 5.155 for x. The second one is out of range, and therefore the solution appears to be x = 2.845. Plotting the original function seems to confirm this.

Thought problem: Would it be easier to write the function for cost as a function of the angle between the ocean line and the shore?