an open rectangular box is to be made from a piece of cardboard 8 inches wide and 8 inches long by cutting a square from each corner and bending up the sides.
a. express the volume of the box as a function of the size x cutout
V = x(8 - 2x)^2
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To express the volume of the box as a function of the size x cutout, we need to determine the dimensions of the box first.
Let's assume the length of the cutout square on each side is x inches. This means that the width of the box would be 8 - 2x inches (as there are two cutouts on the length of the cardboard).
Similarly, the length of the box would be 8 - 2x inches (as there are two cutouts on the width of the cardboard).
The height of the box would be x inches (as the cutout square is bent up to form the height of the box).
Therefore, the volume V of the box can be calculated by multiplying its length, width, and height:
V = (8 - 2x) * (8 - 2x) * x
V = (64 - 16x + 4x^2) * x
V = 4x^3 - 16x^2 + 64x
So, the volume of the box is given by the function V(x) = 4x^3 - 16x^2 + 64x.
To express the volume of the box as a function of the size of the cutout, we need to understand how the dimensions of the box change with respect to the size of the cutout.
Let's assume that the size of the cutout is x inches. This means that a square of size x will be cut from each corner of the cardboard. When we fold up the sides, the dimensions of the resulting box will be as follows:
Length of the box = original length of the cardboard - 2x
Width of the box = original width of the cardboard - 2x
Height of the box = x
The volume of the rectangular box can be calculated by multiplying its length, width, and height. Therefore, the volume of the box as a function of the size of the cutout (x) is:
V(x) = (original length - 2x) * (original width - 2x) * x
In this case, the original length and width of the cardboard are both 8 inches. Substituting these values into the equation, we have:
V(x) = (8 - 2x) * (8 - 2x) * x
Simplifying the equation further, we get:
V(x) = (64 - 16x + 4x^2) * x
Thus, the volume of the box as a function of the size of the cutout (x) is V(x) = 4x^3 - 16x^2 + 64x.