Question

what is the approximate area of the shaded sector in the circle shown below? 70° 7 yd 8.6 yd² 124.0 yd² 29.9 yd² 35.4 yd²

Explanation:

Step1: Find the central angle of the shaded sector

The total angle in a circle is \(360^\circ\). The unshaded sector has a central angle of \(70^\circ\), so the central angle of the shaded sector (\(\theta\)) is \(360^\circ - 70^\circ = 290^\circ\)? Wait, no, wait. Wait, looking at the diagram, the unshaded part is \(70^\circ\), so the shaded part's central angle is \(360 - 70 = 290\)? Wait, no, maybe I misread. Wait, the radius is 7 yd. Wait, the formula for the area of a sector is \(\frac{\theta}{360^\circ} \times \pi r^2\), where \(\theta\) is the central angle of the sector. Wait, maybe the shaded sector is the larger one, so \(\theta = 360 - 70 = 290\)? Wait, no, let's check again. Wait, the unshaded sector is \(70^\circ\), so the shaded sector's angle is \(360 - 70 = 290\) degrees? Wait, but let's calculate the area. Wait, radius \(r = 7\) yd. So first, calculate the area of the whole circle: \(A = \pi r^2 = \pi \times 7^2 = 49\pi\) square yards. Then, the fraction of the circle that is shaded is \(\frac{360 - 70}{360} = \frac{290}{360}\). Then, the area of the shaded sector is \(\frac{290}{360} \times 49\pi\). Let's compute that. First, simplify \(\frac{290}{360} = \frac{29}{36}\). Then, \(49\pi \approx 49 \times 3.1416 \approx 153.938\). Then, \(\frac{29}{36} \times 153.938 \approx \frac{29 \times 153.938}{36}\). Let's calculate 29153.938: 153.93830 = 4618.14, minus 153.938 = 4618.14 - 153.938 = 4464.202. Then, divide by 36: 4464.202 / 36 ≈ 124.0056. So approximately 124.0 yd². Wait, but let's check the steps again.

Wait, maybe I made a mistake in the central angle. Wait, maybe the shaded sector is the one with angle \(360 - 70 = 290\)? Wait, but let's confirm the formula. The area of a sector is \(\frac{\theta}{360} \times \pi r^2\), where \(\theta\) is the central angle of the sector. So if the unshaded sector is \(70^\circ\), then the shaded sector is \(360 - 70 = 290^\circ\). Then, \(r = 7\) yd. So:

Step1: Calculate the area of the whole circle

\(A_{\text{circle}} = \pi r^2 = \pi \times 7^2 = 49\pi \approx 49 \times 3.1416 \approx 153.938\) yd².

Step2: Find the fraction of the circle that is shaded

The central angle of the shaded sector is \(\theta = 360^\circ - 70^\circ = 290^\circ\). So the fraction is \(\frac{\theta}{360^\circ} = \frac{290}{360} = \frac{29}{36}\).

Step3: Calculate the area of the shaded sector

\(A_{\text{sector}} = \frac{29}{36} \times 153.938 \approx \frac{29 \times 153.938}{36}\). Let's compute 29153.938: 153.93820=3078.76, 153.938*9=1385.442, total=3078.76+1385.442=4464.202. Then, 4464.202 / 36 ≈ 124.0056, which is approximately 124.0 yd².

Answer:

124.0 yd²