# 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?

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?
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Hayden Espinoza
First you set up your variables and formulae:
Let r=radius and l=length
Then girth $G=2\pi r$
Volume $V=\pi {r}^{2}l$
We know that $G+l=2\pi r+l\le 108$ as per regulation.
We will take =108 to get the maximum and 'lose' the l
$l=108-2\pi r$
We can now express the volume in r only:
$V=\pi {r}^{2}\cdot l=\pi {r}^{2}\cdot \left(108-2\pi r\right)=108\pi {r}^{2}-2{\pi }^{2}{r}^{3}$
To maximise we set the derivative to 0
$V\prime =216\pi r-6{\pi }^{2}{r}^{2}=6\pi r\cdot \left(36-\pi r\right)=0\to$
$36-\pi r=0\to r=36/\pi \approx 11.46$
Substituting:
$G=2\pi r=72$
$l=108-72=36$
$V=\pi \cdot {11.46}^{2}\cdot 36=14853$