Multiple choice

A piece of paper in the shape of a sector of a circle of radius 10cm and of angle $\displaystyle 216^{\circ}$ just covers the lateral surface of a right circular cone of vertical angle$\displaystyle 2\theta$ .Then $\displaystyle \sin: \theta$ is

  1. $\displaystyle \frac{3}{5}$
  2. $\displaystyle \frac{4}{5}$
  3. $\displaystyle \frac{3}{4}$

  4. none of these
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A Correct answer
Explanation

Sector radius R = 10, angle = 216 degrees. Arc length = (216/360) * 2 * pi * 10 = 12 * pi. This arc becomes the circumference of the cone base: 2 * pi * r = 12 * pi, so r = 6. Slant height l = 10. sin(theta) = r / l = 6 / 10 = 3/5.

AI explanation

The sector forms the lateral surface of the cone, so the arc length of the sector equals the circumference of the cone's base, giving 2 * pi * r = 2 * pi * 10 * (216 / 360) which results in a base radius of r = 6 cm. Since the radius of the sector becomes the slant height of the cone, l = 10 cm. In the right triangle formed by the height, base radius, and slant height, sin(theta) = r / l = 6 / 10 = 3/5. The result is 3/5.

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