A box with a square base and no top is to be made from a square piece of cardboard by cutting 4-in. squares from each corner and folding up the sides, as shown in the figure. The box is to hold 256 in3. What is the length of each side of the square piece of cardboard?
Let the side length of the square piece of cardboard be x inches.
When we cut out 4-inch squares from each corner, the length of each side of the resulting square base will be x - 8 inches (since we cut 4 inches from two adjacent sides).
The height of the box will be 4 inches, since we folded up the sides.
The volume of the box is the product of the length, width, and height, so we have:
(x - 8)(x - 8)(4) = 256
(x - 8)^2 = 16
Taking the square root of both sides, we have:
x - 8 = 4
x = 12
Therefore, the length of each side of the square piece of cardboard is 12 inches.
To find the length of each side of the square piece of cardboard, we can work backwards from the volume of the box.
The formula for the volume of a rectangular prism, which includes a square base, is given by:
Volume = Length x Width x Height
In this case, the length and width are the same since the base is a square, and the height is the length of the sides that are folded up.
Let's call the length of each side of the square piece of cardboard 'x'.
When the corners are cut and the sides are folded up, we are left with a box with dimensions (x-8) x (x-8) x 4.
So, the volume of the box is:
Volume = (x-8) x (x-8) x 4
Given that the volume is 256 in3, we can set up the following equation:
256 = (x-8) x (x-8) x 4
Simplifying the equation, we have:
64 = (x-8) x (x-8)
Taking the square root of both sides, we have:
8 = x-8
Adding 8 to both sides, we get:
x = 16
Therefore, the length of each side of the square piece of cardboard is 16 inches.