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.
520/5 = 104 sq inches for base
Area of cardboard = 105 + 4(5^2)
Length = √area
To find the length of the piece of cardboard needed, we need to understand the dimensions of the tray that will be created.
Let's assume the side length of the square base of the tray is 'x' inches. After cutting 5-inch squares from each corner and folding up the sides, the height of the tray will also be 5 inches.
The length and width of the tray will be reduced by twice the amount of 5 inches (as we are cutting 5 inches from both length and width of the cardboard). Therefore, the length of the tray will be (x - 10) inches, and the width will also be (x - 10) inches.
The formula to find the volume of a rectangular prism is length × width × height. In this case, the volume of the tray is given as 520 cubic inches.
So we have the equation:
(x - 10) × (x - 10) × 5 = 520
Expanding and simplifying the equation, we get:
5(x^2 - 20x + 100) = 520
5x^2 - 100x + 500 = 520
5x^2 - 100x - 20 = 0
Now we can solve this quadratic equation to find the value of 'x'. We can either use the quadratic formula or factorize the equation.
Factoring the equation, we get:
5(x - 2)(x - 2) = 0
From this, we can see that (x - 2) = 0. So, x = 2
Since 'x' represents the side length of the square base, the length of the piece of cardboard needed will be (x + 10) inches.
Substituting the value of 'x', we get:
Length = 2 + 10 = 12
Therefore, the length of the piece of cardboard needed is 12 inches.