Question

a rectangular piece of metal is 10 in. longer than it is wide. squares with sides 2 in. long are cut from the four corners and the flaps are folded upward to form an open box. if the volume of the box is 832 in. cubed, what were the original dimensions of the piece of metal

Answer 1

The problem involves finding the original dimensions of a rectangular piece of metal used to create a box with a known volume by cutting and folding. The metal's length is 10 inches greater than its width, and we must solve an equation based on the volume formula to discover the dimensions.

Step-by-step explanation:

The student is asking about a problem where a rectangular piece of metal is used to form a box by cutting out squares from each corner and folding the flaps upward. The conditions are that the piece of metal is 10 inches longer than its width, and after cutting and folding the corners, the volume of the box is 832 cubic inches. We need to find the dimensions of the original piece of metal.

Let's denote the width of the metal sheet as w inches. Then its length would be w + 10 inches. After cutting 2-inch squares from each corner and folding the flaps to form the box, the dimensions of the box will be w - 4 inches in width, w + 6 inches in length, and 2 inches in height (since the sides of the cut squares are 2 inches).

To find the original dimensions, we use the formula for the volume of a rectangular prism (the box), which is Volume = length × width × height. The volume of our box is given as 832 cubic inches, so we set up the equation (w - 4) × (w + 6) × 2 = 832. By solving this equation, we can find the value of w, and subsequently, the original dimensions of the metal sheet.

Answer 2

volume of the box = length * width * height

length of original peice = x + 10 , width = x

length of box (x + 10) - 4 = x + 6
width of box = x - 4
height of box = 2

so we have

2 * (x -4) * ( x + 6) = 832

2x^2 + 4x - 48 = 832

2x^2 + 4x - 880 = 0
x^2 + 2x - 440 = 0

(x + 22)(x - 20) = 0

x = 20

so width of original piece of metal = 20 ins , length = 30 inches.