A power house, P, is on one bank of a straight river 200 m wide, and a factory, F, is on the opposite bank?

400 m down stream from P. The cable has to be taken across the river under water at a cost of $6/m. On land the cost is $3/m. What path should be chosen so that the cost is minimized?

Let's look at the extreme cases first:

Straight across the river is 200 m @ $6/m = $1200

plus 400 m on land @ $3/m = $1200, total $2400.

If we go diagonal, then by Pythagoras,

it's 200 sqrt(5) m @ $6/m = 1200 sqrt(5) ~= $2683

Let the point straight across from P be P'.

Let's choose a landfall point Q, somewhere P' and F.

Let distance from P' to Q be q.

So total distance is

sqrt (200² + q²) (diagonal under the river) + 400 - q (on land)

cost is 6 * sqrt(40000+q²) + 3 * (400 - q)

we want to minimize that.

first we can divide by 3 -- same value for q will be max

2 * sqrt(40000+q²) + 400 - q

Derivative is

2q / sqrt(40000+q²) - 1

We set that to 0

2q / sqrt(40000+q²) = 1

2q = sqrt(40000+q²)

Square:

4 q² = 40000 + q²

3 q² = 40000

q = 200/sqrt(3)

Length under water is then 400/sqrt(3) = approx 231

and length on land is 400 - 200/sqrt(3) = approx 284

Cost is 6 * 400/sqrt(3) + 3 (400 - 200/sqrt(3)) = 2239.23048

F

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284

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Q

\

|.\

|..\

q..231 (diagonal), q = 116

|.....\

|......\

P'.....P

..200..

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