Check the solutions to the nine circle problems:
1. Circumference Calculation: Solution: C = 2πr = 2π(5) ≈ 31.42 inches.
2. Area Calculation: Solution: A = πr² = π(10/2)² ≈ 78.54 square meters.
3. Arc Length Calculation: Solution: L = (60/360) * 2π(8) ≈ 8.38 centimeters.
4. Sector Area Problem: Solution: A = (45/360) * π(12²) ≈ 56.55 square inches.
5. Tangent Length Problem: Solution: There is no real solution because the point is outside the circle, and the tangent cannot be drawn.
6. Inscribed Angle Problem: Solution: The inscribed angle is half the measure of the intercepted arc, so θ = 0.5 * 60 = 30 degrees.
7. Chord Midpoint Problem: Solution: h = √(6² - (10/2)²) ≈ 3.32 meters.
8. Segment Height Problem: Solution: h = 10 - √(10² - (16/2)²) ≈ 4 inches.
9. Euler's Formula Application: Solution: Using Euler's formula, V - E + F = 8 - 12 + F = 2, we find F = 6. So, the polyhedron has 6 faces.