Question

An open cylindrical tank whose base is a circle is to be constructed of metal sheet so as to contain a volume of πa³ cu cm of water. Find the dimensions so that the quantity of the metal sheet required is minimum.

Answer 1

Let x be the radius of the base, h be the height, V be the volume and S be the total surface area of the cylindrical tank.

∴ by the second derivative test, S is minimum

when x = a 

When x = a, from (1)

\(h = \frac{a^3}{a^2} = a\)

Hence, the quantity of metal sheet is minimum when radius height = a cm.