Multiple choice

From a circle of radius $15\ cm$, a sector central angle $216^{o}$ is cut and its bounding radii are joined without overlap so as to form a cone. Find its volume.

  1. $1081.3\ cm^{3}$
  2. $1071.3\ cm^{3}$
  3. $1018.3\ cm^{3}$
  4. $1061.9\ cm^{3}$
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C Correct answer
Explanation

Sector radius R=15, angle=216 degrees. Arc length = (216/360)*2*pi*15 = 18*pi. This becomes the circumference of the cone base: 2*pi*r = 18*pi, so r=9. Slant height l=15. Height h = sqrt(15^2 - 9^2) = 12. Volume = (1/3)*pi*r^2*h = (1/3)*pi*81*12 = 324*pi = 1017.87. 1018.3 is the closest approximation.

AI explanation

The length of the arc of the sector becomes the circumference of the base of the cone, so 216 divided by 360 multiplied by 2 times pi times 15 gives the base circumference as 18pi cm. This means the base radius r of the cone is 9 cm. The original sector radius becomes the slant height l of the cone, and using the Pythagorean theorem, the height h is the square root of (15 squared minus 9 squared), which equals 12 cm. Applying the cone volume formula, (1/3) times pi times (9 squared) times 12, yields 324pi cubic cm, which is approximately 1018.3 cubic cm.

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