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

• Jun 17th 2012, 09:34 PM
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
• Jun 17th 2012, 10:09 PM
pickslides
Re: Height and radius of cylinder with the least possible surface area given the volu
Therefore $h = \frac{100}{\pi r^2}$ now minimise surface area in terms of height.
• Jun 17th 2012, 11:00 PM
biffboy
Re: Height and radius of cylinder with the least possible surface area given the volu
Quote:

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
• Jun 18th 2012, 10:14 AM
richard1234
Re: Height and radius of cylinder with the least possible surface area given the volu
You can let $h = \frac{100}{\pi r^2}$ and substitute $h$ 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 $\lambda$ such that

$\nabla A = \lambda \nabla V$, where A(r,h) and V(r,h) are the surface area and volume functions, respectively.