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The area $ A $ of the circular dome is:
A = \pi R^2 = \pi \left( rac{s\sqrt{3}}{3}
ight)^2 = \pi \cdot rac{3s^2}{9} = rac{\pi s^2}{3}
Thus, the area of the dome is $ oxed{\dfrac{\pi s^2}{3}} $.
Question: A cartographer uses satellite data to map a circular forest reserve. A straight river crosses the reserve forming a chord of length 14 km, and the shortest distance from the forestâs center to the river is 5 km. What is the area of the forest in square kilometers?
Solution: Let $ r $ be the radius of the forest. The chord length is 14 km, so half is 7 km. The perpendicular distance from the center to the chord is 5 km. Using the Pythagorean Theorem:
r^2 = 5^2 + 7^2 = 25 + 49 = 74
Thus, the area of the forest is:
\pi r^2 = 74\pi
The area is $ oxed{74\pi} $ square kilometers.