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.