An open box is to be made. This box is to have a bottom and four sides, but no top. The box is to be made from paper that is 7" by 12". Maximize the volume and find the height.

To maximize the volume of the box, we need to find the dimensions that will allow us to use the maximum area of the paper.

Let's start by drawing a diagram:

+------------------+
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+------------------+

The bottom of the box will have dimensions x by y, and the height of the box will be h.

The paper has dimensions 7" by 12", so we can think of cutting out squares from each corner of the paper to form the four sides and folding up the remaining flaps to create the box.

Let's say we cut out squares with side length s from each corner:

+---------s--------+
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s | |
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+------------------+

Then the remaining flaps will have the following dimensions:

+----------+----------+
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s x s
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o----------+----------o
y

Now, the volume of the box can be calculated as V = (x - 2s)(y - 2s)h.

To maximize the volume, we need to find the values of x, y, and s that will maximize the expression (x - 2s)(y - 2s)h.

Since we want to use the maximum area of the paper, we need to maximize (x - 2s)(y - 2s). This means x - 2s and y - 2s should be as large as possible.

The maximum value of s can be found by considering the smallest of the two dimensions of the paper (7" and 12") and dividing it by 2. In this case, the smallest dimension is 7", so s = 7/2 = 3.5".

Now, let's substitute some values into our expression for volume to find the height h:

V = (x - 2s)(y - 2s)h

To simplify things, let's substitute (x - 2s) with a, and (y - 2s) with b:

V = abh

Now we can maximize V by considering the dimensions a and b.

Since the box has a bottom and four sides, the width of the paper will become the length of the sides, which is represented by a and b. So, a = 12" and b = 7".

Let's substitute these values into the volumetric equation:

V = 12"h * 7"h
V = 84h^2

To maximize V, we need to find the value of h that maximizes this expression.

Since h is a positive value, the bigger it is, the bigger the volume will be. So, the maximum volume will be achieved when h is at its maximum value.

Therefore, the height h should be as large as possible, which means the height of the box should be h = ∞ (infinity).

However, in practical terms, the height will be limited by the size of the paper, so we need to consider the constraints.

Given that the paper is 7" by 12", the maximum height will be determined by the shorter dimension, which is 7".

So, in this case, the maximum height of the box is 7".

To summarize, to maximize the volume of the box made from a 7" by 12" paper, the height of the box should be 7".