# What maximum volume?

• November 18th 2009, 12:33 PM
mathcalculushelp
What maximum volume?
USPS regulations require that any package must have length plus girth(distance around) of no more tha 84inches.Given this,what are the radius and height of a cylindrical package of maximum volume that is still mailable by parcel post?

I know we need to maximise this.But am stuck.Any suggestion?
• November 18th 2009, 01:12 PM
Scott H
We first state the problem in mathematical terms. Our formula for the volume of a cylinder is

$V=h\cdot\pi r^2.$

To maximize volume, we must have

$h+2\pi r=84$

or

$h=84-2\pi r.$

Substituting this into our formula for $V$, we have

$V=(84-2\pi r)\cdot \pi r^2.$

To find the maximum volume, we note that since $V$ is differentiable everywhere and equals $0$ at the endpoints of its domain, it must attain a maximum at a point where $V'(r)=0$.