a tray with a square base is to be made from a square piece of cardboard by cutting 5 inch squares from each corner and folding up the sides. If the box is to hold a volume 520 cubic inches, find the length of the piece of cardboard that is needed.
To find the length of the piece of cardboard needed, we need to determine the dimensions of the tray, including the length of the side of the square base.
Let's assume that the side length of the square base after the corners are cut is 'x' inches. Then, the dimensions of the tray can be expressed as follows:
Length = x + 2(5 inches) = x + 10 inches
Width = x + 2(5 inches) = x + 10 inches
Height = 5 inches (since it is folded up)
The formula for the volume of a rectangular prism (tray) is given by:
Volume = Length × Width × Height
Substituting the given volume, 520 cubic inches, and the dimensions of the tray, we can set up the equation:
520 = (x + 10) × (x + 10) × 5
Now, let's solve the equation:
520 = 5(x + 10)^2
104 = (x + 10)^2
√104 = x + 10
√104 - 10 = x
x ≈ 7.211
Since x represents the side length of the square base, we need to round it up to the nearest whole number: x = 8 inches.
Therefore, the length of the piece of cardboard needed is the sum of the side lengths of the base plus the doubled amount we cut from each corner, which gives:
Length = x + 2(5 inches) = 8 + 2(5 inches) = 8 + 10 inches = 18 inches
So, the length of the piece of cardboard needed is 18 inches.
To find the length of the piece of cardboard needed to make the tray, we need to determine the dimensions of the tray and then calculate the total length of the cardboard required.
Let's start by understanding the dimensions of the tray. We are given that 5-inch squares are cut from each corner of the cardboard. When these squares are removed and the cardboard is folded up, the sides of the tray will be 5 inches in height.
Let's assume that the side length of the square base of the tray is x inches. After removing the 5-inch squares from each corner, the dimensions of the base will be (x - 2*5) inches, which simplifies to (x - 10) inches.
The height of the tray is given as 5 inches.
The volume of a rectangular prism can be calculated by multiplying its length, width, and height. In this case, the volume is given as 520 cubic inches.
Therefore, the volume of the tray is (x - 10) * (x - 10) * 5 = 520.
Simplifying this equation, we get (x - 10)^2 = 104.
Taking the square root of both sides to isolate x - 10, we find x - 10 = √104.
Adding 10 to both sides gives us x = √104 + 10.
Using a calculator, we find √104 ≈ 10.198.
Therefore, x ≈ 10.198 + 10 ≈ 20.198.
So, the side length of the square base of the tray is approximately 20.198 inches.
Since the height of the tray is 5 inches, the total length of the cardboard needed can be calculated as the sum of the base perimeter and twice the height.
The base perimeter is 4*(x - 10) = 80.792.
Adding twice the height, we get 80.792 + 2*5 = 90.792 inches.
Therefore, the length of the piece of cardboard needed to make the tray is approximately 90.792 inches.