Question

A solid is formed by adjoining one hemisphere to the top of a cylinder. The total volume of the solid is V (treat as a constant) cubic centimeters. Find the minimum surface area of the solid in terms of V.

Answer 1

with radius r and height h,

pi r^2 h + 2/3 pi r^3 = v
so,
h = 3(v - pi r^2)/(2pi r^3)

a = pi r^2 + 2pi r h + 2pi r^2
= 3pi r^2 + 2pi r h
= 3pi r^2 + 3(v - pi r^2)/r^2
= 3pi r^2 - 3pi + 3v/r^2

da/dr = 6pi r - 6v/r^3
dv/dr = 0 when r^4 = v/pi

a(∜(v/pi)) = 3pi√(v/pi) - 3pi + 3√(pi*v)
= 3√pi (2√v - 1)

as usual, double-check the arithmetic and the algebra. I'm pretty sure about the calculus.

Answer 2

thanks a ton.