Question

An open box is to be constructed from a square piece of sheet metal by removing a square of side 4 feet from each corner and turning up the edges. If the box is to hold 576 cubic​ feet, what should be the dimensions of the sheet​ metal?

Answer 1

To find the dimensions of the sheet metal, subtract twice the width of the removed square from the original dimensions. Set up the equation for the volume of the open box. Solve the equation to find the dimensions of the sheet metal.

Step-by-step explanation:

To find the dimensions of the sheet metal, we need to find the dimensions of the open box. The box is created by removing a square of side 4 feet from each corner of the sheet metal and then folding up the edges. Let's assume the side length of the sheet metal is x.

1. The dimensions of the open box can be found by subtracting twice the width of the removed square from the original dimensions of the sheet metal. Since we removed a square of side length 4 feet from each corner, the width of the removed square is 4 feet. So the length of the box is x - 8 feet, and the width of the box is also x - 8 feet.
2. The height of the opened-up edges is 4 feet, since each edge is formed by folding up a side of the removed square.
3. The volume of the box can be found by multiplying the length, width, and height. Since the volume is given as 576 cubic feet, we have the equation (x - 8)(x - 8)(4) = 576.
4. Simplifying the equation, we get (x - 8)(x - 8) = 144.
5. Expanding the equation, we get x^2 - 16x + 64 = 144.
6. Subtracting 144 from both sides, we get x^2 - 16x - 80 = 0.
7. Factoring the quadratic equation, we get (x - 20)(x + 4) = 0.
8. Setting each factor equal to zero, we find the possible values for x: x - 20 = 0 or x + 4 = 0. Solving for x, we get x = 20 or x = -4.
9. Since the side length cannot be negative, we discard the solution x = -4.
10. Therefore, the dimensions of the sheet metal are 20 feet by 20 feet.

Answer 2

20'x20'

Step-by-step explanation:

Given that an open box is to be constructed from a square piece of sheet metal by removing a square of side 4 feet from each corner and turning up the edges

Let a be the side of the square

When 4 feet is cut from each corner, and made into a box the dimensions would be a-8, a-8 and 4.

Volume of box = lwh

`= (a-8) (a-8) 4`

This is equal to 576

`(a-8) (a-8) 4=576\\2(a-8) =24\\a-8 =12\\a=20`

Dimensions of the sheet should be 20x20 feet