A water line runs east-west. A town wants to connect two new housing development

Question

A water line runs east-west. A town wants to connect two new housing developments to the line by running lines from a single point on the existing line to the two developments. The first development is 5 miles south of the existing line; the other development is 6 miles south of the existing line and 3 miles east of the first development. How many miles east from the point due north of the first development should the connection be made to minimize the total length of new line? Please show your work explain your answer

Explanation / Answer

In my diagram L1 and L2 are the lines that you need to minimize so your equation is
y = L1 + L2
using the pythagorean theorem
5^2 + x^2 = L1^2
so L1 = sqrt(x^2 + 25)
same way for L2
6^2 + (3 - x)^2 = L2^2

(3 - x)^2 = x^2 - 16x - 45

so L2 = sqrt(x^2 - 16x - 9)
Your final equation is y = sqrt(x^2 + 25) + sqrt(x^2 - 16x - 9)

If you plug this in on a graphing calculator you will get a nice parabola. Calculating the minimum of that line, I got x = 2.143 miles
That is the distance east from the first development to place on the water line.
The length of the two diagonal lines combined is 8.602 miles.

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