an open box is to be formed by cutting squares from a square sheet of metal that is 10 cm on a side and then folding up the sides. What i the maximum volume of the box?

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To find the maximum volume of the box, we need to determine the size of the squares that need to be cut from the corners of the square metal sheet.

Let's denote the length of the side of the square to be cut from each corner as 'x'. When these squares are cut and the sides are folded up, the resulting shape will be an open box.

To form the box, the height of the box will be equal to the length of the square cut from the corners (x), and the base of the box will be the square formed after cutting the corners, which will have sides of length (10 - 2x). Note that we subtract 2x from the original side length of 10 cm because two corners are being removed.

The volume of the box can be calculated by multiplying the height, base, and width (which is equal to the length of the square metal sheet). In this case, the width remains constant at 10 cm.

So the volume (V) of the box can be expressed as V = x * (10 - 2x) * 10

To maximize the volume, we need to find the value of 'x' that yields the maximum result. To do so, we can take the derivative of the volume equation with respect to 'x' and set it equal to zero:

dV/dx = 0

After finding the critical points, we can determine which one corresponds to the maximum volume. However, for simplicity, we can also solve this problem by examining the endpoints of the interval for 'x'.

Since 'x' represents the size of the squares cut from the corners, it must be greater than zero and less than half the side length of the metal sheet. So, 0 < x < 5.

By evaluating the volume equation for the critical points and endpoints, we can determine the maximum volume.