The Correct Answer and Explanation is:
The correct answer is (192π − 144√3) m².
To find the exact area of the shaded region, we begin with the area of the sector and subtract the area of the triangle formed by the two radii and the chord connecting their endpoints. This shaded region resembles a segment of the circle, not the full sector.
Start by calculating the area of the sector. The formula for the area of a sector is:
Asector=θ360⋅πr2A_{\text{sector}} = \frac{θ}{360} \cdot πr²
Given:
- Radius r=24r = 24 meters
- Central angle θ=120∘θ = 120^\circ
Asector=120360⋅π⋅242=13⋅π⋅576=192π m²A_{\text{sector}} = \frac{120}{360} \cdot π \cdot 24² = \frac{1}{3} \cdot π \cdot 576 = 192π \text{ m²}
Next, calculate the area of the triangle formed by the two radii and the included 120° angle. This triangle is not a right triangle, so we use the sine area formula:
Atriangle=12absin(C)A_{\text{triangle}} = \frac{1}{2} ab \sin(C)
Here, a=b=24a = b = 24, and angle C=120∘C = 120^\circ:
Atriangle=12⋅24⋅24⋅sin(120∘)=12⋅576⋅32=1443 m²A_{\text{triangle}} = \frac{1}{2} \cdot 24 \cdot 24 \cdot \sin(120^\circ) = \frac{1}{2} \cdot 576 \cdot \frac{\sqrt{3}}{2} = 144\sqrt{3} \text{ m²}
Now subtract the area of the triangle from the sector:
Ashaded=192π−1443 m²A_{\text{shaded}} = 192π − 144\sqrt{3} \text{ m²}
This result represents the area of the circular segment bounded by the arc and the chord. It is the exact area of the shaded region shown in the image.
Hence, the correct expression is: (192π − 144√3) m².
