A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 12 cubic centimeters. Find the radius of the cylinder that produces the minimum surface area.

Thank You

Printable View

- Jan 27th 2008, 07:02 AMdoctorgkOptimization
A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 12 cubic centimeters. Find the radius of the cylinder that produces the minimum surface area.

Thank You - Jan 27th 2008, 08:32 AMHenderson
Since [tex]V = \frac {4}{3} \pi r^3 + \pi r^2 h[/maTH],

[tex]12 = \frac {4}{3} \pi r^3 + \pi r^2 h[/maTH] and

$\displaystyle h = \frac {12}{\frac {4}{3} \pi r^3 + \pi r^2}$

Also, $\displaystyle SA = 4 \pi r^2 + 2 \pi rh$, which is

$\displaystyle SA = 4 \pi r^2 + 2 \pi r \frac {12}{\frac {4}{3} \pi r^3 + \pi r^2}$.

Simplify, derive, set equal to zero, and solve.