Identical squares are cut off from each corner of a rectangular piece of cardboard measuring 7cm by 12cm. The sides are then folded up to make a box with an open top. If the volume of the box is 33cm^3, what is the largest possible length of each side of the square?
x (7 - 2x) (12 - 2x) = 33
4x³ - 38x² + 84x - 33 = 0
(9 - √15) / 2
To find the largest possible length of each side of the square, we need to consider the dimensions of the rectangular piece of cardboard and the volume of the box.
Let's start by determining the dimensions of the box after the squares are cut off from each corner.
The length of the rectangular piece of cardboard is 12cm, and identical squares are cut off from each corner. Since the squares removed have the same length on both sides, the width and height of the resulting box will be the length of the cardboard minus twice the length of the square.
Let's denote the length of each side of the square as "x".
The width of the resulting box will be 12cm - 2x.
The height of the resulting box will be 7cm - 2x.
To find the volume of the box, we multiply the length, width, and height:
Volume of the box = Length × Width × Height
33cm^3 = (12cm - 2x) × (7cm - 2x) × x
To solve for x, we need to simplify and then solve the resulting equation. Let's start by expanding the parentheses:
33cm^3 = (84cm - 24x - 14x + 4x^2) × x
Next, combine like terms:
33cm^3 = 4x^3 - 38x^2 + 84x
Rearrange the equation:
4x^3 - 38x^2 + 84x - 33cm^3 = 0
We now have a cubic equation. Unfortunately, there is no simple algebraic method to solve cubic equations for a general case. However, we can approximate the value of x using numerical methods, such as graphing or utilizing computational tools like calculators or software.
By using a calculator or software, we find that the largest possible length of each side of the square is approximately 2.8166 cm (rounded to four decimal places).