The length of a piece of cardboard is two inches more than its width. an open box is formed by cutting out 4 inch squares from each corner and folding the sides. If the volume of the box is 672 cubic inches, find the dimensions.
Volume of rectangular prism
672 = 4*w*(w+2)
Thus:
w^2 + 2w - 168 = 0
Solve for w, using quadratic formula
w = (-b ± √(b^2-4ac))/(2a),
where
a = 1, b = 2, c = -168
To solve this problem, we can start by using the given information to define the dimensions of the piece of cardboard and the dimensions of the box.
Let's assume that the width of the cardboard is "x" inches. According to the problem statement, the length of the cardboard is two inches more than its width, so we can say that the length is "x + 2" inches.
To form the open box, we cut out 4-inch squares from each corner of the cardboard and then fold the sides. This means that the width of the resulting box will be reduced by 8 inches (2 squares on each side, each with a length of 4 inches). So, the width of the box will be "x - 8" inches.
Similarly, the length of the box will be reduced by 8 inches on each side, so the length of the box will be "(x + 2) - 8" inches, which simplifies to "x - 6" inches.
Now, we can calculate the volume of the box using the given information. The volume of a rectangular box is given by the formula:
Volume = Length × Width × Height
In this case, the height of the box is 4 inches (since we cut out 4-inch squares from each corner). Therefore, we have:
672 = (x - 6) × (x - 8) × 4
Now, we can solve this equation to find the value of "x".