Question
A rectangle is placed around a semicircle as shown below. The width of the rectangle is 4 yd. Find the area of the shaded region. Use 3.14 for pie and do not round your answer. Be sure to include the correct unit in your answer
Answers
GPT-4o mini
To find the area of the shaded region, which is the area of the rectangle minus the area of the semicircle, we can follow these steps:
1. **Determine the dimensions of the rectangle**:
- The width of the rectangle is given as \(4\) yards. Since the rectangle is placed around a semicircle, the length of the rectangle is equal to the diameter of the semicircle.
- Therefore, the diameter of the semicircle is \(4\) yards, and the radius \(r\) is given by:
\[
r = \frac{\text{Diameter}}{2} = \frac{4}{2} = 2 \text{ yards}
\]
2. **Calculate the area of the rectangle**:
- The length of the rectangle is equal to the diameter (which is \(4\) yards), and the width is equal to the radius (going from the base of the semicircle to the top), which is also \(2\) yards. So, the area \(A_{\text{rectangle}}\) is:
\[
A_{\text{rectangle}} = \text{Length} \times \text{Width} = 4 \times 2 = 8 \text{ square yards}
\]
3. **Calculate the area of the semicircle**:
- The area of a semicircle \(A_{\text{semicircle}}\) can be found using the formula:
\[
A_{\text{semicircle}} = \frac{1}{2} \pi r^2
\]
- Substituting \(r = 2\) yards and \(\pi \approx 3.14\):
\[
A_{\text{semicircle}} = \frac{1}{2} \times 3.14 \times (2)^2 = \frac{1}{2} \times 3.14 \times 4 = \frac{12.56}{2} = 6.28 \text{ square yards}
\]
4. **Calculate the area of the shaded region**:
- The area of the shaded region is the area of the rectangle minus the area of the semicircle:
\[
A_{\text{shaded}} = A_{\text{rectangle}} - A_{\text{semicircle}} = 8 - 6.28 = 1.72 \text{ square yards}
\]
Thus, the area of the shaded region is \(\boxed{1.72} \text{ square yards}\).
1. **Determine the dimensions of the rectangle**:
- The width of the rectangle is given as \(4\) yards. Since the rectangle is placed around a semicircle, the length of the rectangle is equal to the diameter of the semicircle.
- Therefore, the diameter of the semicircle is \(4\) yards, and the radius \(r\) is given by:
\[
r = \frac{\text{Diameter}}{2} = \frac{4}{2} = 2 \text{ yards}
\]
2. **Calculate the area of the rectangle**:
- The length of the rectangle is equal to the diameter (which is \(4\) yards), and the width is equal to the radius (going from the base of the semicircle to the top), which is also \(2\) yards. So, the area \(A_{\text{rectangle}}\) is:
\[
A_{\text{rectangle}} = \text{Length} \times \text{Width} = 4 \times 2 = 8 \text{ square yards}
\]
3. **Calculate the area of the semicircle**:
- The area of a semicircle \(A_{\text{semicircle}}\) can be found using the formula:
\[
A_{\text{semicircle}} = \frac{1}{2} \pi r^2
\]
- Substituting \(r = 2\) yards and \(\pi \approx 3.14\):
\[
A_{\text{semicircle}} = \frac{1}{2} \times 3.14 \times (2)^2 = \frac{1}{2} \times 3.14 \times 4 = \frac{12.56}{2} = 6.28 \text{ square yards}
\]
4. **Calculate the area of the shaded region**:
- The area of the shaded region is the area of the rectangle minus the area of the semicircle:
\[
A_{\text{shaded}} = A_{\text{rectangle}} - A_{\text{semicircle}} = 8 - 6.28 = 1.72 \text{ square yards}
\]
Thus, the area of the shaded region is \(\boxed{1.72} \text{ square yards}\).