hemispheres attached to a cylinder

Question

A solid is formed by adjoining two hemispheres to the ends of a circular cylinder. The radius of the hemispheres is the same as the radius of the cylinder. The total volume of the solid is 4 cubic inches. What radius should the cylinder be to produce the minimum surface area?

Answer 1

SA = area of hemisphere + area of round surface

SA = πr² + 2πrh

 

 

Volume equation is

 

πr²h + (4/3)πr³ = 4

3πr²h + 4πr³ = 12

 

 

Solve for h from this volume equation. 

 

h = (12 - 4πr³) / 3πr²

 

Then plug it into the SA formula to get the SA formula in terms of r.  Then, set the derivative of SA equal to zero and solve for r.