Question

An oil refinery is located 1 km north of the north bank of a straight river that is 2 km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 9 km east of the refinery. The cost of laying pipe is $500,000/km over land to a point P on the north bank and $1,000,000/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.)
km

Answer 1

Answer:

  7.86 km

Step-by-step explanation:

Let x represent the distance point P lies east of the refinery. (We assume this direction is downriver from the refinery.)

The cost of laying pipe to P from the refinery (in millions of $) will be ...

  0.5√(1² +x²)

The cost of laying pipe under the river from P to the storage facility will be ...

  1.0√(2² +(9-x)²) = √(85 -18x +x²)

We want to minimize the total cost c. That total cost is ...

  c = 0.5√(x² +1) +√(x² -18x +85)

The minimum value is best found using technology. (Differentiating c with respect to x results in a messy radical equation that has no algebraic solution.) A graphing calculator shows it to be at about x ≈ 7.86 km.

Point P should be located about 7.86 km downriver from the refinery.