Answer: 565.92 square inches.
Explanation:
To find the area of the paperboard that remains, we need to subtract the area of the semicircle from the area of the rectangle.
The rectangle has a length of 33 inches and a width of 24 inches, so its area is:
A_rect = length x width
A_rect = 33 in x 24 in
A_rect = 792 sq in
To find the area of the semicircle, we need to first find its radius. The diameter of the semicircle is the same as the width of the rectangle, which is 24 inches. So, the radius is:
r = 1/2 x diameter
r = 1/2 x 24 in
r = 12 in
The area of the semicircle is:
A_semicircle = 1/2 x pi x r^2
A_semicircle = 1/2 x 3.14 x 12^2
A_semicircle = 1/2 x 3.14 x 144
A_semicircle = 226.08 sq in
To find the area of the paperboard that remains, we subtract the area of the semicircle from the area of the rectangle:
A_remaining = A_rect - A_semicircle
A_remaining = 792 sq in - 226.08 sq in
A_remaining = 565.92 sq in
Therefore, the area of the paperboard that remains is 565.92 square inches.