Thread: Height and radius of cylinder with the least possible surface area given the volume

A cylinder has a volume of 100 cubic inches. What is the height and radius of such a cylinder with the least possible surface area?

Constant volume of a cylinder = (pi)r^(2)h

Surface area, including both ends = 2(pi)r^2 + 2(pi)rh

Therefore now minimise surface area in terms of height.

Originally Posted by Roleparadise

A cylinder has a volume of 100 cubic inches. What is the height and radius of such a cylinder with the least possible surface area?

Constant volume of a cylinder = (pi)r^(2)h

Surface area, including both ends = 2(pi)r^2 + 2(pi)rh

(pi)r^2h=100 So h=100/(pi)r^2 Sub this into S getting S=2(pi)r^2+200/r Put derivative=0 to find required r and then h

You can let and substitute into your surface area function and differentiate with respect to r.


Or, if you know how to use Lagrange multipliers, you can use them: there exists a constant such that


, where A(r,h) and V(r,h) are the surface area and volume functions, respectively.