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### Minimizing the Cost of Phone Line Construction

```
Date: 12/3/95 at 20:58:13
From: Anonymous
Subject: optimization-calculus

A telephone company has to run a line from point A on one side of a
river to another point B that is on the other side, 5km down from the
point opposite A.  The river is uniformly 12km wide.  The company can
run the line along the shoreline to a point C and then under the river
to B.  The cost of the line along the shore is \$1000 per km and the cost
under the river is twice as much.  Where should point C be to minimize
the cost?

(ANSWER: x=4*the square root of 3)

B
---------------!
/ !
/   !
/     !
/       !
A--------------!
C

Thank you in advance for your help.
```

```
Date: 1/15/96 at 10:7:49
From: Doctor Ethan
Subject: Re: optimization-calculus

This problem is much like the last one that you sent in.
Here is how I would approach it.

For the sake of the problem I am considering A to be the point 0,0
B to be 5,12 and C to be x,0

The Total cost

TC =  x * \$1000 + Sqrt[(5-x)(12)] * \$2000

Do you see where I got this?

The first part  x * \$1000
is just the cable along the side from A to C

The second half is the cable from C to B

The Sqrt[(5-x)(12)] is just the Pythagorean theorem to find the length
across the river.

Now see if you can find the right spot for C.

Good luck!

-Doctor Ethan,  The Geometry Forum

```
Associated Topics:
High School Calculus

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