1. A rectangular package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section) of 108 inches. Find the dimensions of the package of maximum volume that can be sent. (Assume the cross section is square.)

Answer 1

Wondering if you have a name.

2L+2G=108
L=54-G

volume=LG^2=G^2(54-G)
dv/dg=2G*54-3G^2=0
G=108/3=39
L=54-39=15
check my math

Answer 2

If the package has a combined length and girth of 108 inches. We can determine that:
4x + y = 108
We are also know that the cross section is square, so:
let V=volume
V = x^2y

Let's solve for y in the first equation and plug it into the second one:
y = 108 - 4x
We have
V = x^2(108 - 4x)

V = 108x^2 - 4x^3
In order to solve for the maximum is when V' = 0
V' = 216x - 12x^2 = 0

x(216 - 12x) = 0
Therefore, we have 2 solutions,
first one is x = 0
And this is the one we will work on:
216 - 12x = 0

12x = 216

x = 18
x = 0 is when the package is a minimum, so the maximum occurs when x = 18. Now let's solve for y
Just plug in the value of x in the first equation:
y = 108- 4x = 108- 4(18) = 36

The answers are:
x = 18 inches
y = 36 inches