A solid is formed by two hemispheres to the ends of a right circular cylinder.total volume of the solid is 6 cubic centimeters. Find the radius of the cylinder that produces the minimum surface area.

Question

A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 6 cubic centimeters. Find the radius of the cylinder that produces the minimum surface area. (Round your answer to two decimal places.)

Answer 1

Total Volume = πr²h + (4/3)πr³ = 6

h = (6-(4/3)πr³)/(πr²) = 6/(πr²) - 4r/3

Surface area is the surface area of the cylinder (not counting top and bottom) plus the surface area of a sphere.

A = 2πrh + 4πr²

A ' = 2π(h + rh') + 8πr , h' = -12πr/(πr²)² -4/3

= (-36 + 8πr³)/(3r²) --> Set to zero --> r = (9/(2π))^1/3 ≈1.13

But h = 0 with that value of r. So just the sphere surface area would give the minimum.