Question

A given quantity of metal is to be cast into a solid half circular cylinder with a rectangular base and semi-circular ends. Show that in order that total surface area is minimum, the ratio of length of cylinder to the diameter of semi-circular ends is π ∶ π + 2.

Answer 1

Let r be the radius and ℎ be the height of half cylinder

Volume = 1/2 πr2ℎ = V (constant)……………..(1)

Total surface area of half cylinder is

• = 2(1/2 πr2) + πrh + 2rh .....(2)

From (1) put the value of ℎ in (2)

For maxima/minima ds/dr = 0

V = (π2r3)/(π +2) ....(4)

From (1) and (4)

height : diameter = π : π + 2

Differentiating (3) with respect to r

= positive (as all quantities are +ve)

so * is minimum when

height : diameter = π : π + 2