A rectangular paperboard measuring 33in long and 20in wide has a semicircle cut out of it, as shown below.Find the area of the paperboard that remains. Use the value 3.14 for π, and do not round your answer. Be sure to include the correct unit in your answer.20in33in

Sure, let's solve this step by step.

1. Calculate the area of the rectangular paperboard: The area $A$ of a rectangle is given by the formula: $A = \text{length} \times \text{width}$ Here, the length is 33 inches and the width is 20 inches. So, $A = 33 \, \text{in} \times 20 \, \text{in} = 660 \, \text{square inches}$

2. Calculate the area of the semicircle: The area $A$ of a semicircle is given by the formula: $A = \frac{1}{2} \pi r^2$ To find the radius $r$ of the semicircle, we need to know the diameter. Since the semicircle is cut out from the width of the rectangle, the diameter is equal to the width of the rectangle, which is 20 inches. Therefore, the radius $r$ is: $r = \frac{\text{diameter}}{2} = \frac{20 \, \text{in}}{2} = 10 \, \text{in}$ Now, using $\pi = 3.14$: $A = \frac{1}{2} \times 3.14 \times (10 \, \text{in})^2 = \frac{1}{2} \times 3.14 \times 100 \, \text{in}^2 = 157 \, \text{square inches}$

3. Calculate the remaining area of the paperboard: The remaining area is the area of the rectangle minus the area of the semicircle: $\text{Remaining area} = \text{Area of rectangle} - \text{Area of semicircle}$ Substituting the values we found: $\text{Remaining area} = 660 \, \text{square inches} - 157 \, \text{square inches} = 503 \, \text{square inches}$

So, the area of the paperboard that remains is $503 \, \text{square inches}$.