Expert: Scotto - 1/1/2011

Question
Can you answer this question for me?
I can't illustrate this problem. There are some information that I can't understand. please help me.

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 10km from Town B. if the points 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?

Answer
It states that the two towns are 20 km apart on the same side of the river.
It also states that the river is 15 km from A and 10 km from B.

The best place to put this pumping station would be on the same side of the river
and the x-axis as the river.  It will be x km up the river, so it is at (x,0).
Town A will be at (0,15), and town B at (20,10).

This will make the distance to town A be sqrt(x^2 + 15^2) and
the distance to town B be sqrt((20-x)^2 + 10^2).

Now 15^2 = 225 and 10^2 = 100, so the distances are sqrt(x^2 + 225) and sqrt((x-20)^2 + 100)

To find the total distance, add these two together, getting
y = sqrt(x^2 + 225) + sqrt[(20-x)^2 + 100] where y is the total distance.

To minimize it, take the derivative of y and set it equal to 0.
The derivative would be dy/dx = 2x/[2*sqrt(x^2 + 225)] - 2(20-x)/{2*sqrt[(20-x)^2 + 100]}.
There is a 2 in the exponet from the x term being squared a 2 in the denominator from the function being to the 1/2, and these will cancel, leaving
dy/dx = x/sqrt(x^2 + 225) - (20-x)/sqrt[(20-x)^2 + 100].

If we set this to 0, we get 0 = x/sqrt(x^2 + 225) - (20-x)/sqrt[(20-x)^2 + 100].
Adding the 2nd term of the two terms to both sides gives
(20-x)/sqrt[(20-x)^2 + 100] = x/sqrt(x^2 + 225).

Squaring both sides gives (20-x)^2/[(20-x)^2 + 100] = x^2/(x^2 + 225).

To finis this problem, cross multiply.
This gives (20-x)^2 * (x^2 + 225) = x^2 * [(20-x)^2 + 100].

Now (20-x)^2 = 400 - 40x + x^2, so we have
(400 - 40x + x^2)(x^2 + 225) = x^2(400 - 40x + x^2 + 100).

Muliply this out and both sides will have a -40x^3 + x^4,
so the answer comes down to subtracting one side from the other and then
solving a quadratic equation.