An open-topped box can be created by cutting congruent squares from each of the four corners of a piece of cardboard that's has dimensions of 20cm by 30cm and folding uo the sides. Determine the dimensions of the squares that must be cut to create a box with a volume of 1008cm³
Let the side length of the squares cut from each corner be x.
After cutting the squares and folding up the sides, the dimensions of the resulting box will be as follows:
Length: (30 - 2x) cm
Width: (20 - 2x) cm
Height: x cm
Given that the volume of the box is 1008 cm³, we can set up the equation:
Volume = Length x Width x Height
1008 = (30 - 2x)(20 - 2x)(x)
1008 = 600x - 100x² + 40x - 4x²
1008 = -100x² - 4x² + 600x + 40x
1008 = -104x² + 640x
104x² - 640x + 1008 = 0
Now we need to solve for x. This is a quadratic equation that can be solved by factoring, completing the square, or using the quadratic formula.
After solving for x, we can find the dimensions of the squares by substituting back into the expressions for the Length and Width:
Length = 30 - 2x
Width = 20 - 2x
Therefore, the dimensions of the squares that must be cut to create a box with a volume of 1008 cm³ will be the value of x in cm.