Question

What are the radius, length and volume of the largest cylindrical package that may be sent using a parcel delivery service that will deliver a package only if the length plus the girth (distance around) does not exceed 108 inches?

Answer 1

First you set up your variables and formulae:

Let `r=`radius and `l=`length

Then girth `G=2pir`
Volume `V=pir^2l`

We know that `G+l=2pir+l<=108` as per regulation.
We will take `=108` to get the maximum and 'lose' the `l`
`l=108-2pir`

We can now express the volume in `r` only:
`V=pir^2*l=pir^2*(108-2pir)=108pir^2-2pi^2r^3`

To maximise we set the derivative to `0`

`V'=216pir-6pi^2r^2=6pir*(36-pir)=0 ->`
`36-pir=0->r=36//pi~~11.46`

Substituting:
`G=2pir=72`
`l=108-72=36`
`V=pi*11.46^2*36=14853`