A solid is formed by attaching a hemisphere to each end of a cylinder. If the total volume is to be 120cm^3, find the radius (in cm) of the cylinder that produces the minimum surface area. Express your answer correct to 2 decimal places. Help!?

1 Answer
Aug 14, 2017

r = ((3V)/(4pi))^(1/3).

Explanation:

Assume without loss of generality the cylinder has length l, the hemispheres and cylinder have radius r, and the entire geometric object has volume V and surface area S.

Two hemispheres attached to either end have the equivalent volume of a single sphere, 4/3 pi r^3. The cylinder has volume pi r^2 l.

Then we write,

V = 4/3 pi r^3 + pi r^2 l,
l = V/(pi r^2) - 4/3 r.

The surface area of the geometric object will be the surface area of a sphere with radius r, 4 pi r^2, plus the curved surface area of a cylinder with radius r and length l, 2 pi r l.

So we write,

S=2 pi r l + 4 pi r^2.

Substituting the definition of l in terms of r and V gives,

S = 2V/r - 8/3 pi r^2 + 4 pi r^2,
S = 2Vr^(-1)+4/3pi r^2.

Given that V is a fixed positive number, this function is increasing as r -> + infty and goes to +infty as r -> 0. Then it turns and obtains a minimum value as required in this interval.

We solve for the turning points by differentiating and equating with zero to find the value(s) of r for which the function turns.

("d"S)/("d"r) = 8/3 pi r - 2V/r^2,

Setting ("d"S)/("d"r)=0 and solving for r gives r^3 = (3V)/(4pi) which has one real positive root for r, r = ((3V)/(4pi))^(1/3).