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A circle with radius 3 has a sector with a central angle of (11)/(15)pi radians. What is the area of the sector?

A circle with radius 33 has a sector with a central angle of 1115π \frac{11}{15} \pi radians. What is the area of the sector?

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Full solution

Q. A circle with radius 33 has a sector with a central angle of 1115π \frac{11}{15} \pi radians. What is the area of the sector?
  1. Use Formula: To find the area of the sector, we need to use the formula for the area of a sector, which is (θ/2π)∗πr2(\theta/2\pi) * \pi r^2, where θ\theta is the central angle in radians and rr is the radius of the circle.
  2. Plug in Values: First, let's plug in the values we know: θ=(1115)π\theta = \left(\frac{11}{15}\right)\pi and r=3r = 3. So, the area of the sector is ((1115)π2π)∗π∗32\left(\frac{\left(\frac{11}{15}\right)\pi}{2\pi}\right) * \pi * 3^2.
  3. Simplify Equation: Simplify the equation by canceling out π\pi in the numerator and denominator. We get 1115/2×9\frac{11}{15} / 2 \times 9.
  4. Multiply to Find Area: Now, multiply (1115)(\frac{11}{15}) by 92\frac{9}{2} to find the area.\newline(1115)×(92)=9930.(\frac{11}{15}) \times (\frac{9}{2}) = \frac{99}{30}.
  5. Final Answer: Simplify 99/3099/30 to get the final answer.\newline99/3099/30 simplifies to 3.33.3, but wait, that's not right because we can't have a decimal in a fraction simplification.

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