What is the optimal package volume for UPS?

[bowniski]

Homework Statement

UPS will only accept packages with a length of no more than 108 inches and length plus girth of no more than 165 inches. Assumign that the front face of the package is square, what is the largest volume package that UPS will accept?

Assuming the package looks like this,
http://imgur.com/4FeLW

Homework Equations

Girth = perimeter of the face (4x)

The Attempt at a Solution

Okay, so I got these two equations:

Volume: V = (x²)(y)
y $$\leq$$ 108
y + 4x $$\leq$$ 165

How do I solve this from here? I tried to solve for X by moving 4x to the other side, then setting them equal but it didnt seem to work!

Thanks!

 

[Outlined]

Next step is to change the second line to y + 4x = 165 (together with the obvious constraint 0 =< 4x =< 165).

 

[bowniski]

Next step is to change the second line to y + 4x = 165 (together with the obvious constraint 0 =< 4x =< 165).

Thanks for the reply!

Okay, why do you assume that y+4x = 165? Is it because that's the highest value possible?

In any case, if I do that:

y+4x = 165
y = 165-4x

v = (x^2)(165-4x)
v = 165x^2 - 4x^3
v' = -12x^2 + 165x

Did I do this right? If so, should I just find the maximum of v' and that would be my answer for x?

With that, the maximum is 55/8 = x

 

[Outlined]

there is an error in v', you missed 2 * 165

Think yourself about why it is smart to put y + 4x = 165. Would a solution which does NOT hit the 165 be an optimal solution (as there is room for x and y to be bigger, as you did not hit the 165)??

 

[bowniski]

there is an error in v', you missed 2 * 165

Think yourself about why it is smart to put y + 4x = 165. Would a solution which does NOT hit the 165 be an optimal solution (as there is room for x and y to be bigger, as you did not hit the 165)??

Oh true, that was silly of me to ask! Haha! Thanks for the correction, I didn't see that.