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.
2007-12-13 17:52:16 · 2 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
Actually, a sphere has max volume for a given surface area, so the cylinder will have height = 0.
V = (4/3)pi*r^3+pi*r^2*h = 12 ... sphere + cylinder
h = [12 - (4/3)*pi*r^3]/(pi*r^2)
Surface area s(r) = 4pi*r^2 + 2pi*r*h ... sphere + cylinder
= 4pi*r^2 + [24/r - (8/3)pi*r^2] = (4/3)pi*r^2 + 24/r
s'(r) = (8/3)pi*r - 24/r^2 = 0
r^3 = 9/pi, so r = (9/pi)^1/3
h = [12 - (4/3)*pi*r^3]/(pi*r^2)
= [12 - (4/3)*pi*(9/pi)]/(pi*r^2) = 0, as expected
2007-12-13 18:20:20 · answer #1 · answered by halac 4 · 0⤊ 0⤋
3-X to the square rooot of....
Oh forget it... I'm too drunk
2007-12-14 01:59:57 · answer #2 · answered by box of rain 7 · 0⤊ 2⤋