An open tank is to be constructed with square base and vertical sides so as to contain a given quantity of water. Show that the expenses of lining with lead will be least, if depth is made half of width.

Explanation

Let the length, width and height of the open tank be `x, x` and `y` with respectively. Then its volume is `{x}^{2} {y}` and total surface area is `{x}^{2}+4 {xy}`. It is given that the tank can be hold a given quantity of water. This mean that its volume is constant. Let it be `{V}`.


`V=x^{2} y`



The cost of the material will be least if the total surface area is least. Let's denote total surface area


`{S}={x}^{2}+4 {xy}`


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