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 8 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?

Answer 1

Let x = no of km downriver to P and C = cost.
10√(x^2+ 1) + 5×(8- x) = C
5(2x)÷√(x^2 + 1) - 5 = 0
10x = 5√(x^2 + 1)
100x^2 = 25(x^2 + 1)
⇔75x^2 = 25
⇒ x=√(25÷75)

x ≈ √3/3 ( Let x = no of km downriver to P and C = cost. )