The US Postal service stipulates that any boxes sent through the mail must have a length plus girth totaling no more than 108 inches. Find the dimensions of the box with maximum volume that can be sent through the us mail assuming that the width and height of the box are equal

Question

The US Postal service stipulates that any boxes sent through the mail must have a length plus girth totaling no more than 108 inches.  Find the dimensions of the box with maximum volume that can be sent through the us mail assuming that the width and height of the box are equal

anonymous · Answer

currently studying applications of extrema if that helps

anonymous · Answer

Your best bet for this would be a cube, so$$108=5s$$Which would give us a cube a little more than 21 inches on each side.

anonymous · Answer

Sorry, jumped to the answer by knowing about how these things work.  Let's consider how we will measure this box.

anonymous · Answer

Since the box is square in two dimensions, let the width and height be x.  The length will be 108-4x (this comes from the specification of length plus girth).

anonymous · Answer

So the volume is$$V(x)=x^2(108-4x)=108x^2-4x^3$$

anonymous · Answer

Are you with me so far?

anonymous · Answer

yep, do you find the critical point of that then?

anonymous · Answer

Exactly.

anonymous · Answer

sweet, thank you very much

anonymous · Answer

I'm interested to see if my farmer's estimate is correct.  So we differentiate$$V'(x)=216x-12x^2=0\rightarrow x=0, x=18$$

anonymous · Answer

I think you probably get the maximum at x=18.  I wonder why it's different from my initial estimate?

anonymous · Answer

My calculus answer seems ok, but isn't even a maximum box size.  Something is wrong.

anonymous · Answer

I found another solution online that shows the length being 108-x.  This gives the equation $$V(x)=x ^{2}(108-x)$$ and the derivative of 216x-3x^2.  X would then be 72 inches

anonymous · Answer

making y=36

anonymous · Answer

Yeah, but the girth is the distance around.  If we have a box that has a square base with side length s, the girth is 4s.  That's what needs to be added to the height to get 108

anonymous · Answer

ah ok, yeh you are right then

anonymous · Answer

But my calculus answer is still wrong.

anonymous · Answer

So w = h =18

and l = 108 - 4h = 108 - 4*18 = 108 - 72 = 36

So maximum volume = lwh = 36*18*18 = 11664 cubic inches

anonymous · Answer

Ok, I guess there is something I missed on my initial estimate that it had to be a cube....

anonymous · Answer

Actually, the answer is two cubes stuck end-to-end.  Hmmm.....

anonymous · Answer

Nice work, Astro.  I hope I was helpful.