# help!!!- optimization problem

• Dec 13th 2008, 04:21 PM
52090
help!!!- optimization problem
I dont know how to start on this probem. It would nice if someone helps me.

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 developments. One is 3 miles south of the existing line; the other is 4 miles south of the existing line and 5 miles east of the first development. Find the place on the existing line, relative to the 2 developments, to make the connection that minimizes the total length of the new line.

• Dec 13th 2008, 04:26 PM
Jhevon
Quote:

Originally Posted by 52090
I dont know how to start on this probem. It would nice if someone helps me.

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 developments. One is 3 miles south of the existing line; the other is 4 miles south of the existing line and 5 miles east of the first development. Find the place on the existing line, relative to the 2 developments, to make the connection that minimizes the total length of the new line.

By Pythagoras' theorem, the length of the line connecting the town 3 miles south of the existing line is $\displaystyle \sqrt{x^2 + 3^2}$ and the length of the line connecting the town 4 miles south of the existing line is $\displaystyle \sqrt{4^2 + (5 - x)^2}$. you want to minimize the sum of those

hint: the square roots aren't needed, hope you know why
• Dec 13th 2008, 04:59 PM
Soroban
Hello, 52090!

If we must use Calculus, the approach and set-up is quite ugly.

Quote:

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 developments.
One is 3 miles north of the existing line; the other is 4 miles north of the existing line
and 5 miles east of the first development.

Find the place on the existing line, relative to the two developments,
to make the connection that minimizes the total length of the new line.

Code:

                          * B                         * |     A *              *  |       | *          *    | 4     3 |  *      *      |       |    *  *        |       * - - - * - - - - - *       C  x  P    5-x    D

Development $\displaystyle A$ is 3 miles from the line: .$\displaystyle AC = 3$
Development $\displaystyle B$ is 4 miles from the line: .$\displaystyle BD = 4$
$\displaystyle CD = 5$

Let $\displaystyle P$ be a point on $\displaystyle CD$.
Let $\displaystyle x \,=\, CP$, then $\displaystyle 5-x \,=\, PD$

We want to minimize the distance: $\displaystyle AP + PB$

In right triangle $\displaystyle ACP\!:\;AP \:=\:\sqrt{x^2+3^2}$
In right triangle $\displaystyle BDP\!:\;PB \:=\:\sqrt{(5-x)^2 + 4^2}$

The total distance is: .$\displaystyle D(x) \;=\;\left(x^2+9\right)^{\frac{1}{2}} + \left(x^2-10x+41\right)^{\frac{1}{2}}$

. . and that is the function we must minimize.

I'll wait in the car . . .
.