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A power house, P, is on one bank of a straight river $200$ m wide, and a factory, F, is on the opposite bank $400m$ downstream from P. The cable has to be taken across the river, under water at a cost of Rs $6/m$. On land the cost is Rs $ 3/m$. What path should be chosen so that the cost is minimized?

I first looked at the extreme cases and then tried to find the solution. But then too there was a problem

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Draw a picture. Let $A$ be the point directly across the river from $P$.

Suppose we go underwater directly to a point $x$ metres downstream from $P$. Then by the Pythagorean Theorem, we will have a $\sqrt{x^2+200^2}$ underwater stretch. Then we need to travel $400-x$ overland to $F$. The cost $C(x)$ is given by $$C(x)=6\sqrt{x^2+200^2}+3(400-x).$$ To minimize, find the derivative $C'(x)$ and set it equal to $0$. We have $$C'(x)=\frac{6x}{\sqrt{x^2+200^2}}-3.$$ Solving $C'(x)=0$ is easy. We get $2x=\sqrt{x^2+200^2}$. Square both sides.

When you have done this, and found a candidate $x$, you need to check whether this really gives the minimum cost.

Remark: There are various other ways to solve the problem. We chose the most basic beginning calculus approach.

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