Multiple choice

A sector of circle of radius 10 cm has the angle 120o. It is rolled up so that two bounding radii are joined together to form a cone. Find the volume of the cone.

  1. 109.3 cm3

  2. 107.32 cm3

  3. 105.63 cm3

  4. 108.45 cm3

  5. 102.5 cm2
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A Correct answer
Explanation

Radius of sector R = 10. Arc length L = (120/360) * 2 * pi * 10 = 20pi/3. This arc becomes the circumference of the cone base: 2 * pi * r = 20pi/3, so r = 10/3. Slant height l = R = 10. Height h = sqrt(l^2 - r^2) = sqrt(100 - 100/9) = sqrt(800/9) = 20*sqrt(2)/3. Volume = (1/3) * pi * r^2 * h = (1/3) * pi * (100/9) * (20*sqrt(2)/3) = 2000*pi*sqrt(2) / 81 = 109.3.

AI explanation

When the sector forming a 120 degree angle is rolled into a cone, its radius becomes the slant height of the cone, so l = 10 cm, and the arc length becomes the cone's circumference, meaning 2 * pi * r = (120 / 360) * 2 * pi * 10, yielding a base radius r = 10 / 3 cm. Using the Pythagorean theorem, the height of the cone is the square root of (10^2 - (10 / 3)^2), which simplifies to 10 times the square root of 8 divided by 3, or (20 times the square root of 2) / 3 cm. Applying the volume formula (1 / 3) * pi * r^2 * h, we calculate (1 / 3) * (22 / 7) * (100 / 9) * ((20 * 1.414) / 3) to be approximately 109.3 cubic cm.

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