A sheet metal fabricator makes boxes from sheets of steel. Each sheet of steel has an area of 1500 cm^2. 6 cm squares are cut from each corner and the edges are bent up to form a box with a volume of 3456 cm^2. What are the dimensions of each box?
the area of the bottom of the box is
... 3456 / 6 = 576
(x + 12) [(576 / x) + 12] = 1500
x is one dimension, 576/x is the other
To find the dimensions of the box made from the sheet of steel, we need to follow these steps:
1. Let's assume that the length of the original sheet is "x" cm and the width is "y" cm.
2. We are given that each corner has a 6 cm square cut out. So, when the corners are cut out, the length of the sheet will be reduced by 12 cm (6 cm on each end), and the width will also be reduced by 12 cm.
3. After cutting out the corners, the length of the sheet becomes (x - 12) cm, and the width becomes (y - 12) cm.
4. The edges of the sheet are bent up to form a box. When the sheet is folded, the height of the box will be 6 cm.
5. The volume of the box is given as 3456 cm³. The volume of a rectangular box can be calculated by multiplying its length, width, and height. So, we have the equation:
(x - 12) * (y - 12) * 6 = 3456
6. Simplifying the equation, we get:
(x - 12) * (y - 12) = 576
7. Now, we need to find two numbers, (x - 12) and (y - 12), whose product is equal to 576. We can factorize 576 and find the suitable pair of factors.
The factors of 576 are:
1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, 288, 384, 576
We need to look for two factors whose difference is 12 (which equals to the width and length reductions). From the list, we can see that 24 and 12 meet our requirements.
Therefore, (x - 12) = 24 and (y - 12) = 12.
8. Solving for x and y:
x = 24 + 12 = 36 cm
y = 12 + 12 = 24 cm
So, the dimensions of the box made from the sheet of steel are 36 cm in length, 24 cm in width, and 6 cm in height.