Finding Efficient Shape to Maximize Volume

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mathnerd

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Hey guys, I need help in solving this question. If you could please start me off with some hints, I will try solving it. Thanks.

An open-topped storage box is to have a square base and vertical sides. If 108 m^2 of sheet metal is available for its construction, find the most efficient shape to maximize it's volume.

Please help out on this hard calculus question. Thank you.
 
mathnerd said:
An open-topped storage box is to have a square base and vertical sides. If 108 m^2 of sheet metal is available for its construction, find the most efficient shape to maximize it's volume.
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What have you tried? How far did you get? Where are you stuck? You draw the box, picked a variable for the side-lengths of the base, labelled the picture, wrote down the formulas for volume and surface area, and... then what?

Please be complete. Thank you! :D

Eliz.
 
Sorry. Here is what I did:

[attachment=0:2dqkb8xf]image92.GIF[/attachment:2dqkb8xf]

And then, the side length of the square base is with the variable 's'. I have labeled the picture.

The formula for the area of a cube is s^2
The formula for area of rectangle is a*b

But, I don't understand the fact that the vertical sides are rectangles or cubes too.

The volume of a cube is s^3
The volume of a rectangular prism is V =lwh

Surface area of a cube is 6a^2
Surface area of a rectangular prism is 2(wh+lw+lh).

I think it would be helpful for me, maybe if you could explain the question to me. The question is too vague. Also, I think 108 m^2 is the surface area, but I don't know how to apply it to get maximum volume.

Please help. Thanks.
 

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Thanks Dr. Flim-Flam.

Here is what I did.

s^2 + 4sy = 108
y = (108-s^2) / 4s

plug this "y" into volume formula

V = s^2 (y)
= s^2 ((108-s^2) / 4s)
= ( 108s^2 - s^4 ) / 4s
108s^2 = s^4
108 = s^2
s = 108^0.5

Is this right? Should I continue working this out, meaning now that I have found s, I can find y, Then plug the y and x into volume formula to get the maximized volume?

But, I don't get why the question says "the most efficient shape". Am I on the right track. Please help. Thanks.
 
Thanks very much for your help. After differentiating V(s), and putting V(s)` = 0, the value of s is 6 m

Then, I found the value of y to be 3 m.

The volume comes out to be 108 m^3 (somehow it's the same as the area)

So, is this the answer? Because the question says 'the most efficient' shape. Could I get some more help on that? Thanks.
 
lol. Thanks very much. You are a great helper. Doing this question was fun. Thanks.
 
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