I have to find out which shape of packaging for a fragile object has the most volume to fit the object and styrofoam packing. The sum of the height and the perimeter must be less than $100cm$.
There is a circular base (cylindrical package), a square base (rectangular package) and a rectangle base ($w = 2l$)(rectangular package).
The problem is I have no idea how to start. will the function represent the volume? the base? the height and perimeter? can I assume the y-axis is the volume?
Another question is: what is the perimeter of a cylinder? some one said (not on this site) that it was $2{\pi}r+2h$ but wouldn't it be $4{\pi}r+2h$ because there are two circumferences in a cylinder.
EDIT: this probably is a cubic function that is representing the volume
There is no maximum volume, since the problem specifies that the sum must be less than $100$. So we assume that the regulations say $\le 100$.
To maximize volume, it is best to use the full $100$. For height $h$ and radius $r$, we have the constraint $h+2\pi r=100$. This is because the perimeter of the base is $2\pi r$.
The volume is $\pi r^2 h$. So we want to maximize $\pi r^2(100-2\pi r)$, where $0\le r\le \frac{100}{2\pi}$. To solve this problem, we can use the usual tools (derivative).
If we solve the problem of the previous paragraph, then whatever maximum volume $V$ we get, we can get arbitrarily close to $V$ if we take the less than seriously, but we cannot reach $V$.
The square base and rectangular base problems are handled in a similar way.