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Bot GPT 3.5 To find the exact area of the circle in which the minor segment subtends a central angle of 5π÷6 radians, we first need to calculate the area of the whole circle and then subtract the area of the major segment.
Given that the diameter of the circle is 14 cm, the radius (r) of the circle is half of the diameter, which is 7 cm.
The formula for the area of a circle is A = πr^2.
Therefore, the area of the whole circle is:
A = π(7)^2
A = 49π
Next, we need to find the area of the major segment. Since the central angle of the minor segment is 5π÷6 radians, the central angle of the major segment is 2π - 5π÷6 = 7π÷6 radians.
The formula for the area of a segment of a circle is A = (1/2)r^2(Θ - sinΘ), where r is the radius and Θ is the central angle in radians.
Therefore, the area of the major segment is:
A_major = (1/2)(7)^2(7π÷6 - sin(7π÷6))
A_major = (1/2)(49)(7π÷6 - 1/2)
A_major = (1/2)(49)(7π÷6 - 1/2)
A_major = (49/2)(7π÷6 - 1/2)
A_major = (343π÷12 - 49/2)
Now, to find the area of the minor segment, we subtract the area of the major segment from the area of the whole circle:
A_minor = 49π - (343π÷12 - 49/2)
A_minor = 49π - 343π÷12 + 49/2
A_minor = 588π÷12 - 343π÷12 + 49/2
A_minor = 245π÷12 + 49/2
Therefore, the exact area of the minor segment of the circle with a diameter of 14 cm and a central angle of 5π÷6 radians is 245π÷12 + 49/2 square centimeters.