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Bot GPT 3.5 To find the area of the major segment with a minor segment subtending an angle of 5π/6 radians, we first need to find the area of the sector formed by the minor segment.
The area of a sector of a circle is given by the formula A = (θ/2) * r^2, where θ is the central angle in radians and r is the radius of the circle.
Here, the central angle is 5π/6 radians and the radius is half of the diameter, so the radius (r) is 7 cm.
So, the area of the sector formed by the minor segment is:
A = (5π/6 * 7^2)
A = (35π/3) cm²
Now, we need to find the area of the triangle formed by the minor segment. To do this, we need to find the height of the triangle. The height can be found using the formula h = r * sin(θ/2), where r is the radius and θ is the central angle in radians.
h = 7 * sin(5π/6 / 2)
h = 7 * sin(5π/12)
h = 7 * (√3/2)
h = (7√3)/2
The area of the triangle formed by the minor segment is:
A = (1/2) * base * height
A = (1/2) * 14 * (7√3)/2
A = 49√3 cm²
Finally, the area of the major segment can be found by subtracting the area of the triangle from the area of the sector:
Area of major segment = Area of sector - Area of triangle
Area of major segment = (35π/3) - 49√3
Area of major segment ≈ 31.73 cm²
Therefore, the area of the major segment of the circle is approximately 31.73 cm².