Question

A country's postal service will accept a package if its length plus its girth (the distance all the way around) does not exceed 90 inches. find the dimensions and volume of the largest package with a square base that can be mailed.

Answer 1

Answers:
Dimensions of the package:
length = 32 in
width = height = 16 in
Volume of package = 8192 in³

Explanation:
The attached image shows the shape of the package with a square base.

Part (a): getting the dimensions:
Assume that the length is y in and that width = height = x in (since the base is a square)
This means that the girth is 4*x = 4x in

We are given that the maximum sum of length and girth is 96.
This means that:
y + 4x = 96
This can be rewritten as:
y = 96 - 4x ..............> I

The volume of the package can be calculated as follows:
Volume = length * width * height = y * x * x = x²y in³

Substitute with equation I in the equation of the volume as follows:
Volume = x² (96-4x) = 96x² - 4x³ in³

For the volume to be maximum:
dV / dx = 0
This means that:
2(96) x - 4(3) x² = 0
192x - 12x² = 0
x (192-12x) = 0
either x = 0 .......> This is rejected as the side cannot be zero
or 192 - 12x = 0 ..........> 192 = 12x ..........> x = 16 in ..........> accepted

Substitute with the value of x in equation I to get y as follows:
y = 96 - 4x = 96 - 4(16) = 96 - 64 = 32 in

Part (b): getting the volume:
volume of package = length * width * height 
volume = 32 * 16 * 16
volume = 8192 in³

Hope this helps :)