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furqannoor · Nov 6, 2009, 09:52 PM

It is required to lay a cable from a point A on one side of the bank of a river 100m wide to a point B on the opposite bank 1500m downstream from the point C directly opposite A.(Assuming the river bank are parallel).What rout should be followed to minimize
The cost,if it costs $30.00 per meter to lay cable on land and $ 60.00 per meter to lay in the water?

galactus · Nov 7, 2009, 04:22 AM

Problems very similar to this one can be found by googling. This is a very cliché calculus problem. One can find it in some form or another online or in any calc book.

First, draw a picture. Note, there is a right triangle which the hypoteneuse represents the length of cable under water across the river. See it?

Let the portion along the bank, on land, be 1500-x.

The cost on land is 30(1500-x) and the cost underwater is $60\sqrt{x^{2}+100^{2}}$

Then, the total cable cost will be $C=30(1500-x)+60\sqrt{x^{2}+100^{2}}$

This is what must be optimized to find the length across land and under the water. Differentiate, set to 0, and solve for x. Look at the diagram I have provided. See what is going on?