Find the dimensions of the box made from square 52cm on side

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beachbunny619

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An open box with a square base is to be made from a square piece of cardboard 52 centimeters on a side by cutting out a square from each corner and turning up the sides.

Find the dimensions of the box that yield the maximum volume
 
Re: Find the dimensions of the box?

Hello, beach:

Please read the post entitled "Read Before Posting," which is available in all the forums. Then come back and show your work so far with this. What are your thoughts? Where are you stuck? From the little bit you have typed above, we have no idea how to help you, because we don't know what part of solving the problem you don't understand.

-Paul
 
Re: Find the dimensions of the box?

Did you make a diagram?. That is always a good idea.

If we let the corners being cut out have length x, then the area of a corner cut out is x^2. See?. Right?.

The length and width of the sheet is 52 cm. Suppose now we cut out the four corners. What is the length of a side now?. Wouldn't it be 52-2x?.

Can you continue?. After we fold up the sides, what would be the depth of the box?. Can you see?.

Get a piece of paper and cut out the corners, then fold it up. What is its depth?.
 
Re: Find the dimensions of the box?

No, I am sorry, that is incorrect. Remember, only one of your solutions will be in the domain. That is, one will be equal to or larger than half the width of the box. If the solution is, say, 26 then if you have just cut the sheet in 4 pieces.

Your solution of 9 is very close though. 36 is not a viable solution. It is too big. See why?.
 
Re: Find the dimensions of the box?

9 is the size of the square your are cutting out(though, not correct, but close). That is, it's the length and width dimension of the cut out square that results in the largest box in volume.

You know volume is length*width*height. What is the length after you cut out the corners?. What is the width after you cut out the corners?. What is the depth after you cut our the corners?.

Show me what you come up with for your volume equation so I can see what you are doing. Okey-doke?.
 
Re: Find the dimensions of the box?

No, not quite.

The depth of the box would be x, wouldn't it?. Because that is the dimension of our corner we cut out.

The width and length will be the same because it's a square: 52-2x

Now, volume is \(\displaystyle V=\underbrace{x}_{\text{depth}}\cdot \overbrace{(52-2x)}^{\text{width}}\cdot \underbrace{(52-2x)}_{\text{length}}\). See why?.

If we expand this out we get a cubic which is easily differentiated to a quadratic and easy to solve. Remember, you will get two solutions but only one will be good.

Let me know what you get.
 
Re: Find the dimensions of the box?

No need. Let's muddle through it and you can show your teacher you got the correct solution. I want you to see why, though.

We now have the volume formula. Differentiate it, set to 0 and solve for x. You were close with your first solution of 9.

The formula expands to \(\displaystyle V=4x^{3}-208x^{2}+2704x\)

You can differentiate that one, can't you?. Polynomials are among the easiest to differntiate.
 
Re: Find the dimensions of the box?

galactus said:
No need. Let's muddle through it and you can show your teacher you got the correct solution. I want you to see why, though.

We now have the volume formula. Differentiate it, set to 0 and solve for x. You were close with your first solution of 9.

The formula expands to \(\displaystyle V=4x^{3}-208x^{2}+2704x\)

You can differentiate that one, can't you?. Polynomials are among the easiest to differntiate.
Click to expand...

So it goes 12x^2- 416x+ 2704?
 
Re: Find the dimensions of the box?

Yep, now set that to 0 and solve. It's just a quadratic. Use the quad formula if you have trouble factoring it. You can factor out a 4 though, then solve the quad in the parentheses.
 
Re: Find the dimensions of the box?

Very good. I suppose you had the answer then to check?.
 
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