An open box is made from a square piece of metal by cutting out a 4 inch square from each corner and turning up the sides. Find the area of the original square if the box is to contain:
A) 16 cubic inches
And
B) 400 cubic inches
let the piece of metal be x inches by x inches
base of box is (x-8) by (x-8) , where x
so
4(x-8)(x-8) = 16
x^2 - 16x + 64 = 4
(x-6)(x-10) = 0
x = 6 or x = 10 , but remember x > 8
so the original piece of metal was 10 inches by 10 inches
You try B
To find the area of the original square, we need to use the formula for the volume of a rectangular prism, which is length × width × height. In this case, since the box is made from a square piece of metal, the length and width will be equal.
A) To find the area of the original square if the box is to contain 16 cubic inches, we need to determine the height of the box. Let's denote the length of one side of the metal piece as x.
When you cut out a 4-inch square from each corner and fold up the sides, the height of the box will be 4 inches. Thus, the length and width of the base of the box will be x - 8 inches (since 4 inches are removed from both sides).
The volume of the box is given by (x - 8) × (x - 8) × 4 = 16 cubic inches.
Simplifying this equation, we have (x - 8)^2 = 4.
Taking the square root of both sides, we get x - 8 = 2.
Solving for x, we have x = 10.
Therefore, the length of one side of the original square is 10 inches, so the area of the square is 10 × 10 = 100 square inches.
B) To find the area of the original square if the box is to contain 400 cubic inches, we repeat the process.
The volume of the box is given by (x - 8) × (x - 8) × 4 = 400 cubic inches.
Simplifying this equation, we have (x - 8)^2 = 100.
Taking the square root of both sides, we get x - 8 = 10.
Solving for x, we have x = 18.
Therefore, the length of one side of the original square is 18 inches, so the area of the square is 18 × 18 = 324 square inches.