A box with a square base and no top is to be made from a square piece of cardboard by cutting 6 in. squares from each corner and folding up the sides. The box is to hold 11094 in cubed. How big a piece of cardboard is needed?
___in. by ___ in.
To determine the size of the cardboard needed, we need to find the dimensions of the square base and the height of the box.
Let's start by understanding the process of creating the box using the given information:
1. We need to cut 6-inch squares from each corner of the cardboard.
2. Then, we fold up the sides to form the walls of the box.
Let's assume the side length of the square base is "x" inches. Since we are cutting 6-inch squares from each corner, the dimensions of the square base will be reduced by 12 inches (6 inches from each side).
Therefore, after we cut out the squares and fold up the sides, the base of the box will have a side length of (x - 12) inches.
The height of the box will be equal to the length of the 6-inch square cuts we made at each corner. Hence, the height will also be 6 inches.
To calculate the volume of the box, we multiply the dimensions:
Volume of the box = (x - 12) inches * (x - 12) inches * 6 inches
According to the given information, the box should hold 11094 cubic inches. So we can set up the equation:
(x - 12) inches * (x - 12) inches * 6 inches = 11094 in^3
Now, we can solve this equation to find the value of x, which will give us the side length of the cardboard needed.
11094 in^3 = (x - 12)^2 * 6 in
Divide both sides of the equation by 6:
1849 in^3 = (x - 12)^2
Now, take the square root of both sides:
√(1849 in^3) = x - 12
√(43^2 in^2) = x - 12
43 in = x - 12
Add 12 to both sides to solve for x:
x = 43 in + 12 in
x = 55 in
So, the side length of the cardboard needed is 55 inches.
Therefore, the piece of cardboard should be:
55 in. by 55 in.