A rectangle is three times as long as it is wide. Squares of side 2 feet are cut from each corner. Then the sides are folded up to make an open box. Express the volume of the box as a function of the width, x.
box width ... x - (2 * 2)
box length ... 3 x - (2 * 2)
box depth ... 2
v(x) = 2 (x - 4) (3 x - 4)
Thanks Scott!
To express the volume of the box as a function of the width, x, we first need to find the dimensions of the box after the squares are cut from the corners and the sides are folded up.
Let's start by defining the width of the rectangle as x. Since the length is three times the width, the length of the rectangle would be 3x.
When squares of side 2 feet are cut from each corner, the dimensions of the rectangle will decrease by 4 feet (2 feet on each side). So, the new width and length of the rectangle after the squares are cut would be (x - 4) and (3x - 4) respectively.
Now, let's consider folding up the sides to form the box. The height of the box will be the length of the original square that was cut from the corners, which is 2 feet.
Therefore, the volume of the box can be calculated by multiplying the length, width, and height of the box, i.e., (x - 4) * (3x - 4) * 2.
Hence, the volume of the box as a function of the width, x, is given by the expression:
V(x) = 2(3x - 4)(x - 4).