Help with optimization question

Question

An oil refinery is located 1 km north of the north bank of a straight river that is 1 km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 8 km east of the refinery. The cost of laying pipe is $500,000 per km over land to a point P on the north bank and $1,000,000 per km under the river to the tanks. To minimize the cost of the pipeline, how far downriver from the refinery should the point P be located? (Round the answer to two decimal places.)

Answer 1

So the first step in this problem would be to draw a picture. You should include the tanks, the refinery, and point P. Let the south bank of the river lie on the x-axis, put the tanks at the origin, and put the refinery at point (-8,2). Point P should be at (x,1) where we are looking for x. You can just put it where you think it would be for now. This graph is not for accuracy it is to help us create the cost of function.

The pipe will consist of two straight sections. The first section will go from the tanks to P, and the second will go from P to the refinery. Using the distance formula,

l_TP = √(x² + 1)

l_RP = √((-8-x)²+1)

where l_TP is the length of the pipe from the tanks to P and l_RP is the length from the refinery to P.

The cost function is a function of x given as,

c(x) = -5×10⁵ √((-8-x)²+1) + 10⁶ √(x² + 1)

where each term is the cost of the section of pipe.

To minimize cost we take the derivative and set it equal to zero.

dc/dx = 5×10⁵ (x-8)/√((-8-x)²+1) + 10⁶ x/√(x² + 1) = 0

This does not look like it has an easy solution so you might want to graph dc/dx and use a calculator to find its intercept.

x ≈-0.57

Make sure to double check my algebra!