Question

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!?

Answer 1

`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)`.