Problem of finding the volume of a box.

[naaa00]

Homework Statement

The airlines accept a box if length + width + height <= 62. If h is fixed, show that the maximum volume (62-w-h)wh is V = h(31 - h/2)^2.

The Attempt at a Solution

Well, I multipied:

V(h) = (62-w-h)wh
V(h) = 62wh - w^2h - wh^2

dv/dt = 62w - w^2 - 2w = 60w - w^2

0 = 60w - w^2
w = 60.

But this can not be. This is wrong. I tried other things but all of my attemps are wrong...

 

[HallsofIvy]

What you have done makes no sense at all. You write "V(h)" when V depends on both w andh as variables. And then you write "dv/dt" when there is NO "t" variable at all!

A differentiable function of two variables will have max or min only where its gradient is 0 or, equivalently, \partial V/\partial w= 0 and \partial V/\partial h= 0.

 

[naaa00]

Thank you for your answer, HallsofIvy.

"dv/dt" well, yes. That was a typo from my part. My bad. Clearly volume does not depent on t...

I see that you have written partial derivative notation. I am assuming that your suggestion is to use partials, since volume seems to depend on two variables, namely h and w. But this exercise is in a problem set of a single-variable calculus textbook. I guess this can be solved that way, but for the moment I think the textbook expects me to solve the problem without the aid of partials. (I don't know know anything of partials. I only know something about its notation.)

My thought is that since V = HLW, I would get something of the form V = h times M^2, if I assume L = W = M^2, and then substitute. But I don't know how to prove this assumption and I have the feeling that it is wrong...

 

[naaa00]

But the problem statement says h is fixed. I am not sure what does the textbook means with "fixed" (I am not an english native speaker.) I suppose it means that h is not a variable, since it is "fixed"; it does not change. If that is the case, then V depends on L and W. But the problem ask me to show that (62-w-h)wh is V = h(31 - h/2)^2. Here "V" seems to be in function of h; a some sort of contradiction. I am a little bit confused...

 

[naaa00]

Well, certainly I was confused yesterday. Happily now I undestand the problem:

V = (62 - h - w)wh,

V = 64wh - h^2w - w^2h.

dv/dw = 62h - h^2 - 2wh,

0 = 62h - h^2 - 2wh,

w = 62h - h^2/ 2h

w = 31 - h/2.

It turns out that L = W, because L = 62 - h - w, or L = 62 - h - (31 - h/2), or L = 31 - h/2.

Then, if we let W^2 = LW, it is clear that V = W^2H. Finally we have (31 - h/2)^2 times h.

Thanks! See you next time!