Another maximum volume word problem (length + girth)

[Linty Fresh]

A parcel delivery service will deliver a package only if the length plus girth (distance around) does not exceed 108 inches. Find the dimensions of a rectangular box with square ends that satisfies the delivery service's restriction and has maximum volume. What is the maximum volume?

OK, I started with Perimeter P=2x+2y and Volume V=lwh:

108=2x+2y
y=(108-2x)/2=54-x

But I couldn't figure out a way to calculate the squares cut out at the end. So I started again:

x=length of side cut out
length=L-2x
width=W-2x

P=2L+2W=2(L-2x)+2(W-2x)
108=2L-4x+2W-4x
54=L+W-4x

And I get stuck here. I'm sure the answer involves either substituting the perimeter equation into the volume equation or vice-versa, but I can't figure out how to wind up with one variable. Am I on the right track, or do I need to go back to square one?

Thanks so much!

 

[skeeter]

length plus girth (distance around) does not exceed 108 inches.

for max V ... length + girth = 108

L + 2(x + y) = 108

since the end is square ...

L + 2(x + x) = L + 4x = 108

so ... L = 108 - 4x

V = L*x²

V = (108 - 4x)x²

can you now find the value of x that will maximize the volume?

 

[Linty Fresh]

Wow, thanks for your quick response.

OK, I'm still not clear on how you got to this part:

L + 2(x + y) = 108

Why can 2(x+y) fill in for girth? I'm a bit confused as to what (x+y) stands for. It seems by the rest of the steps that it refers to the square you cut out of the ends, but what about the rest of the width?

Thanks for your help! Sorry I'm still a bit slow!

 

[stapel]

Why can 2(x+y) fill in for girth?

What is the definition of "girth"? For what do "x" and "y" stand? :wink:

Eliz.

 

[skeeter]

where are you getting the idea that something is to be "cut out" ?

the package is already made ... it's a rectangular prism with a square base.

the girth of the package is the perimeter of that square base.

the perimeter of a square base of side length x is 4x.

the L is the length of the package, then L + 4x < 108 according to the rules of the delivery service.

 

[Linty Fresh]

I think I might be misreading the problem. I'm a bit confused over the precise definition of a "rectangular box with square ends." I took "square ends" to mean "square ends cut out of each corner to fold the box up" which is probably incorrect. I'm going to go draw a couple of pictures and see if I can figure it out.

Thanks very much for your help, skeeter and stapel.