Here's how we can find the arc length, sector area, and minor segment area for the pendulum's swing:
**1. Arc Length (PQ):**
* We know the angle (θ) is 20° and needs to be converted to radians: 20° * (π/180) = π/9 radians.
* We know the radius (r) is 40 cm.
* Formula for arc length (s) is: s = r * θ
* Therefore, s = 40 cm * (π/9) ≈ 13.96 cm.
**2. Sector Area (OPQ):**
* Formula for sector area (A) is: A = (θ/2) * r^2
* A = (π/9) * (40 cm)^2 ≈ 55.85 cm^2
**3. Minor Segment Area (PQO):**
* To find the minor segment area, we need to subtract the area of the triangle OPQ from the sector area.
* First, find the area of the triangle OPQ. We know it's an isosceles triangle with a 20° angle at the apex. This means the other two angles are each (180° - 20°)/2 = 80°.
* Using trigonometry, we can find the height of the triangle. Let's say the base of the triangle is 2x. Then, sin(80°) = x/40 cm. Solving for x, we get x ≈ 39.07 cm. Therefore, the height of the triangle is 2x * sin(80°) ≈ 67.82 cm.
* Area of triangle OPQ = (1/2) * base * height = (1/2) * 2x * 67.82 cm ≈ 2662.4 cm^2
* Finally, minor segment area = sector area - triangle area ≈ 55.85 cm^2 - 2662.4 cm^2 ≈ -2606.55 cm^2
**Note:** The minor segment area is negative because the area of the triangle is larger than the sector area. This is because the angle is relatively small, and the triangle takes up a significant portion of the sector.
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