Postal Regulations

See attachment. I am skipping part (b).

Part (a)

Volume = length • width • height

V = y • x • x

V = x^2 y

V(x) = x^2 y

I'm not sure about the domain.

Part (c)

The perimeter of the cross section is
P = x + x + x + x + y.

In short, P = 4x + y.

Let P = 108 and solve for y.

108 = 4x + y

108 - 4x = y

Plug 108 - 4x into V(x) for x.

V(x) = x^2 (108 - 4x)

V(x) = 108x^2 - 4x^3

Factor

V(x) = 4x^2(27 - x)

Let V(x) = 0

0 = 4x^2(27 - x)

Set each factor to 0 and solve for x, where x represent the dimension that will maximize the volume.

It is clear that 4x^2 = 0 leads to 0.
So, x = 0 cannot be the answer.

I know 27 - x = 0 leads to 27.

I am going to say x = 27 is the only dimension
for this application that will maximize the volume.

You say?

 

see attached

 

I thank you. Can the maximum be found without calculus? This question is from a precalculus textbook.

 

maximum and minimum (or critical points) are points where the function is defined and its derivative is zero or undefined

so, you have to use derivative to find max

or you can find it from the graph (see attached)
when you use

https://www.symbolab.com/solver/step-by-step/V=108x^{2}-4x^{3}

scroll down to see graph, you will see a point where max is, put mouse on it and the coordinates (18,11664) will pop out

 

How is my effort?