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 x and h be the base and depth of the tank with volume v

∴v=`x^2` h

⇒h= `(v^2)/x`

surface area,

s=`x^2 +4xh`

Let Q be the total cost and P be the cost per square unit of material

∴Q=`(x^2 +4xh)`P=`(x^2 + (4v)/x )`P

differentiating the above function, we get,
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