A box with a square base and no top is to be made from a square piece of carboard by cutting 8 in. squares from each corner and folding up the sides. The box is to hold 7200 in. How big a piece of cardboard is needed?
To find out how big a piece of cardboard is needed, we first need to determine the dimensions of the square base and the height of the box.
Let's assume that the length of each side of the square base is x inches.
When we cut 8 inches squares from each corner, we are essentially reducing the length and width of the square base by 16 inches (8 inches from each side).
Therefore, the dimensions of the square base after cutting the corners will be (x - 16) inches by (x - 16) inches.
To determine the height of the box, we can look at the folding up of the sides. Each side is folded up, which adds the same length as the depth of the box. In this case, we are given that the box is to hold 7200 in². Since the box has a square base, the volume of the box is equal to the square of the length of the base multiplied by the height of the box:
(x - 16)² * (x - 16) = 7200
To solve this equation, we can start by simplifying it:
(x - 16)(x - 16)(x - 16) = 7200
(x - 16)³ = 7200
Taking the cube root of both sides:
x - 16 = ∛7200
To find the cube root of 7200, we can use a calculator or approximation methods. The cube root of 7200 is approximately 19.6.
Simplifying further:
x = ∛7200 + 16
x ≈ 19.6 + 16
x ≈ 35.6
So, the length of each side of the square base is approximately 35.6 inches.
Now, to find the dimensions of the piece of cardboard needed, we add back the 16 inches that were cut from each side:
Length: x + 16 = 35.6 + 16 = 51.6 inches
Width: x + 16 = 35.6 + 16 = 51.6 inches
Therefore, a square piece of cardboard measuring approximately 51.6 inches by 51.6 inches is needed.
To determine the size of the cardboard needed, we first need to find the dimensions of the square base and the height of the box.
Let:
x = length of the side of the square base (in inches)
h = height of the box (in inches)
Since 8-inch squares are cut from each corner, the side length of the base will be reduced by twice that amount (16 inches) in both dimensions. Therefore, the dimensions of the base will be (x-16) inches by (x-16) inches.
When folding up the sides, the height of the box will be the same as the length of the square cut from the corners, which is 8 inches.
Given that the volume of a rectangular prism is given by V = length x width x height, we can determine the equation for the volume of the box:
7200 in³ = (x - 16) in * (x - 16) in * 8 in
Simplifying the equation, we have:
7200 in³ = (x - 16)² * 8 in
To solve for x, we can start by dividing both sides of the equation by 8:
900 in² = (x - 16)²
Next, take the square root of both sides:
√(900 in²) = x - 16
Simplifying further:
30 in = x - 16
Finally, solve for x by adding 16 to both sides:
x = 30 in + 16 in
x = 46 in
Therefore, a piece of cardboard with dimensions 46 inches by 46 inches is needed to make the box.