mr. m has an 8 inch by 11 inch sheet of paper. He created a box by cutting congruent squares from each corner.The base of his box is 54 square inches.How much did he cut from each corner.
9 * 6 = 54
11 - 9 = 2 (1 cut from each corner)
8 - 6 = 2 (1 cut from each corner)
To find out how much Mr. M cut from each corner, we need to follow these steps:
1. Calculate the area of the base of the box: 54 square inches.
2. Determine the dimensions of the base of the box. Since Mr. M cut congruent squares from each corner, the length and width of the base of the box will be reduced by twice the length of the square he cut.
Let's assume that Mr. M cut x inches from each corner.
The length of the base of the box (after cutting) will be 11 inches - 2x inches.
The width of the base of the box (after cutting) will be 8 inches - 2x inches.
3. To find the area of the base of the box, we multiply the length by the width:
(11 - 2x) inches * (8 - 2x) inches = 54 square inches.
4. Simplify the equation:
(11 - 2x)(8 - 2x) = 54.
Expanding the equation: 88 - 22x - 16x + 4x^2 = 54.
Rearranging the terms: 4x^2 - 38x + 88 = 54.
Subtracting 54 from both sides: 4x^2 - 38x + 34 = 0.
5. Now, we can solve this quadratic equation to find the value of x, which represents how much Mr. M cut from each corner.
This equation can be factored or solved using the quadratic formula.
Applying the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).
In this case: a = 4, b = -38, and c = 34.
Plugging in the values: x = (-(-38) ± √((-38)^2 - 4 * 4 * 34)) / (2 * 4).
Simplifying further: x = (38 ± √(1444 - 544)) / 8.
Continuing to simplify: x = (38 ± √900) / 8.
The two possible values for x are obtained by removing the square root: x1 = (38 + 30) / 8 = 8 and x2 = (38 - 30) / 8 = 1.
6. Since Mr. M cut squares from each corner, we can conclude that he cut 8 inches from each corner of the 8 inch side, and 1 inch from each corner of the 11 inch side.