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

Problems very similar to this one can be found by googling. This is a very cliche 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

Then, the total cable cost will be

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?.

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