Multiple choice

A sector of a circle of radius $10$ cm is folded such that it forms into a cone. If the central angle of the sector is $\displaystyle 144^{^{\circ}}$ then what is the volume of the cone formed ? (in $\displaystyle cm^{3}$ )

  1. $\displaystyle \frac{704\sqrt{2}}{21}$
  2. $\displaystyle \frac{628\sqrt{11}}{11}$
  3. $\displaystyle \frac{576\sqrt{21}}{21}$
  4. $\displaystyle \frac{682\sqrt{11}}{11}$
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A Correct answer
AI explanation

The radius of the sector, 10 cm, becomes the slant height of the cone, and the arc length, calculated as (144/360) times 2 times (22/7) times 10, equals the cone's base circumference to give a radius of 4 cm. Using the Pythagorean theorem, the height of the cone is the square root of 10 squared minus 4 squared, which is 2 times the square root of 21. The volume of the cone, (1/3)(pi)(r^2)(h), becomes (1/3) times (22/7) times 16 times 2 times the square root of 21, which simplifies to 704 times the square root of 21 divided by 21 cubic cm.

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