Question
A container open at the top is a right circular cylinder having a surface (side and lower base) area of 108 square centimeters. What should the radius and altitude be in order to provide the largest possible volume.
Answers
oobleck
so you have
πr^2 + 2πrh = 108
so h = (108-πr^2)/(2πr)
Now, the volume is
v = πr^2 h = πr^2 (108-πr^2)/(2π) = 54r^2 - π/2 r^4
so max vomume occurs when
dv/dr = 0
dv/dr = 108r - 2πr^3 = 2r(54-πr^2)
dv/dr=0 at r=√(54/π) = 3√(6/π)
That makes the height h = (108-πr^2)/(2πr) = 27/(πr)
πr^2 + 2πrh = 108
so h = (108-πr^2)/(2πr)
Now, the volume is
v = πr^2 h = πr^2 (108-πr^2)/(2π) = 54r^2 - π/2 r^4
so max vomume occurs when
dv/dr = 0
dv/dr = 108r - 2πr^3 = 2r(54-πr^2)
dv/dr=0 at r=√(54/π) = 3√(6/π)
That makes the height h = (108-πr^2)/(2πr) = 27/(πr)