What is the radius that will produce 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 2 cubic centimeters. Find the radius of the cylinder that produces the minimum surface area. (Round your answer to two decimal places.)

______cm

*please have answer in cm, thanks

Answer 1

Volume:

4/3 * pi * R^3 + pi * R^2 * H = 2

pi * r^2 * h = 2 - 4/3 * pi * R^3

h = (2 - 4/3 * pi * R^3) / (pi * r^2 )

= 2/( pi * r^2) - (4/3) * r <--- please label this equation ALPHA

Surface Area:

4 * pi * r^2 + 2 * pi * r * h + 2 * pi * r^2 = SA ( r, h)

4 * pi * r^2 + 4/r + 2 * pi * r^2 = SA(r) <--- substitutes equation ALPHA

0 = d SA/ dR = 8 * pi * r - 4 * r^(-2) + 4 * pi * r

0 = 8 * pi * r^3 - 4 + 4 * pi * r^3 = 12 * pi * r^3 - 4

1/(3 * pi) = r^3

r = [1/ (3 * pi) ]^(1/3) = 0.473416028204.....