Apiece of cardboard is 27 inches by 54 inches. a square is to be cut from each corner and the sides folded up to make an open-top box. what is the maximum possible volume of the box? round your answer to the nearest four decimal places.

The volume of a box is the amount of space in it.

The maximum value of the box is

The dimension of the cardboard is given as:

Assume the cut-out is x.

So, the dimension of the box is:

The volume of the box is:

Expand

Differentiate

Set to 0

Rewrite as:

Using a calculator, we have:

21.29 is greater than the dimensions of the box.

So, the possible value of x is:

Recall that:

So, we have:

Hence, the maximum value of the box is

Read more about volumes at:

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Let's call the side of the square

When we cut out the square from each corner, the base of the box is

and the height of the box is of course so the volume is

Approximately, x≈5.70577 or x≈21.294

x can't be bigger than half of 27, so we reject the second value.

We check the second derivative; a negative second derivative indicates a local maximum.

Good. For our answer we want

3787.9951 cubic inches