1. ## Optimization largest volume

Postal regulations specify that a parcel sent by parcel post may have a combined length and girth of no more than 120 in. Find the dimensions of a rectangular package that has a square cross section and the largest volume that may be sent through the mail.
(Hint: The length plus the girth is 4x+h )

2. Originally Posted by tim22
Postal regulations specify that a parcel sent by parcel post may have a combined length and girth of no more than 120 in. Find the dimensions of a rectangular package that has a square cross section and the largest volume that may be sent through the mail.
(Hint: The length plus the girth is 4x+h )
Here the side length of the end is $x$, and the length is $h$.

Thus, the volume is given by: $V = x^2h$

now our constraint is that the length plus girth must be less than or equal to 120. thus we must have that

$4x + h \le 120$

now solve for $h$, we get:

$h \le 120 - 4x$

we have the largest volume when h is as large as possible, thus take h to be such that $h = 120 - 4x$

thus, plugging this into our volume equation, we get:

$V = x^2 (120 - 4x)$

$\Rightarrow V = 120x^2 - 4x^3$

now this last function is what we want to maximize. can you continue?

3. yes i figured it out, thank you very much

4. Originally Posted by tim22
yes i figured it out, thank you very much
you're welcome