An oil refinery is located 1 km north of the north bank of a straight river that
ID: 1302002 • Letter: A
Question
An oil refinery is located 1 km north of the north bank of a straight river that is 3 km wide. 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. To minimize the cost of the pipeline, how far downriver from the refinery should the point P be located? (Round your answer to two decimal places.) Not even sure where to start :/
Explanation / Answer
Suppose that we construct a pipeline across the shore that x km away from the refinery. By the Pythagorean Theorem, we see that the length of pipe across the river is:
L(pipe,river) = ?(x^2 + 3^2) = ?(x^2 + 9).
Then, the length of the remaining pipe across the bank is:
L(pipe,bank) = 8 - x.
The total cost of the pipeline is:
C = 800000*L(pipe, river) + 400000*L(pipe, bank)
= 800000?(x^2 + 9) + 400000(7 - x).
By differentiating:
dC/dx = 800000x/?(x^2 + 9) - 4000000
= 400000[2x - ?(x^2 + 9)]/?(x^2 + 9), by combining fractions.
By setting dC/dx = 0:
2x - ?(x^2 + 9) = 0 ==> x = ?3.
You can show that this produces a minimum by showing that d^2C/dx^2 > 0.
Therefore, the point P should be located ?3 ? 1.73 km away from the refinery.