Question

a parallelogram is cut out of a 12-inch by 8-inch sheet of paper. there are four right triangle remnants. two have the dimensions 2 inches by 9 inches, and the other two have the dimensions 3 inches by 6 inches. the resulting parallelogram has a base of approximately 9.22 inches. complete the following steps to calculate the altitude of the parallelogram using area methods. the area of the sheet of paper is \square square inches. the combined area of the triangle cutouts is \square square inches. the area of the parallelogram is \square square inches. the altitude of the parallelogram rounded to two decimals is \square square inches.

Explanation:

Step1: Calculate the area of the paper

The paper is a rectangle with length 12 inches and width 8 inches. The area of a rectangle is given by \( A = l \times w \).
\( A = 12 \times 8 = 96 \) square inches.

Step2: Calculate the area of each type of triangle

For the triangles with dimensions 2 inches by 9 inches: The area of a right triangle is \( \frac{1}{2} \times base \times height \). So, the area of one such triangle is \( \frac{1}{2} \times 2 \times 9 = 9 \) square inches. There are two of these, so their total area is \( 2 \times 9 = 18 \) square inches.

For the triangles with dimensions 3 inches by 6 inches: The area of one such triangle is \( \frac{1}{2} \times 3 \times 6 = 9 \) square inches. There are two of these, so their total area is \( 2 \times 9 = 18 \) square inches.

Step3: Calculate the combined area of the triangles

The combined area of all the triangle cutouts is \( 18 + 18 = 36 \) square inches.

Step4: Calculate the area of the parallelogram

The area of the parallelogram is the area of the paper minus the area of the triangles. So, \( 96 - 36 = 60 \) square inches.

Step5: Calculate the altitude of the parallelogram

The area of a parallelogram is given by \( A = base \times altitude \). We know the area \( A = 60 \) square inches and the base is approximately 9.22 inches. So, we solve for altitude: \( altitude = \frac{A}{base} = \frac{60}{9.22} \approx 6.51 \) inches.

Answer:

The area of the sheet of paper is \(\boxed{96}\) square inches.
The combined area of the triangle cutouts is \(\boxed{36}\) square inches.
The area of the parallelogram is \(\boxed{60}\) square inches.
The altitude of the parallelogram rounded to two decimals is \(\boxed{6.51}\) inches.