Question

Two towns A and B are to get their water supply from the same pumping station to be located on the bank of a straight river that is 15 km from town A and 10 km from town B. If the points on the river nearest to A and B are 20 km apart and A and B are on the same side of the river, where should the pumping station be located so that the least amount of piping is required.

Let M be location and L be amount of piping

L= AM + BM
Found AM^2 and BM^2
Stuck after that

Answer 1

Assuming that the pipes are to be straight from M to A and B (that is, the pipe does not go to some other point P and then divide), then let's add a couple more points to the diagram:

P: the point on the river bank closest to A
Q: the point on the river bank closest to B

Then PM+MQ = √(20^2-5^2) = √395

Then if M is at distance x from P, it is √395-x from Q.

The amount of piping is AM+BM where
AM^2 = x^2+15^2
BM^2 = (√395-x)^2 + 10^2

So, we have

L = √(x^2+225) + √(x^2-2x√395+495)

L is a minimum when x = 3√(79/5) = 11.92

Not sure what analytical tools you have at your disposal to solve this kind of problem.

Answer 2

Thanks so much. May I know what are the in-between steps for the following two lines please?

L = √(x^2+225) + √(x^2-2x√395+495)

L is a minimum when x = 3√(79/5) = 11.92