Question

Of all the closed cylindrical cans (right circular), which enclose a given volume of 100 cm³, which has the minimum surface area ?

Answer 1

Let the radius and height of right circular cylinder be r and h respectively. 

Given : 

Volume of Cylindrical can = 100 cm³ 

Volume of a cylinder = πr²h 

⇒ πr²h = 100 … (1) 

Surface of a cylinder, 

S = 2πrh + 2πr² 

From equation (1) we get,

⇒ S = 2πr\((\frac{100}{\pi r^2})\) + 2πr²

⇒ S = \(\frac{200}{r}\) + 2πr²

Condition for maxima and minima,

⇒ \(\frac{dS}{dr}\) = 0

This is the condition for minima 

From equation 1,

h = \(\frac{100}{\pi r^2}\)

Hence, required dimensions of cylinders are radius = \((\frac{50}{\pi})^{\frac{1}{3}}\) and height = 2\((\frac{50}{\pi})^{\frac{1}{3}}\)