Multiple choice

The height of a cone and the radius of its base are respectively 9 and 3 cm. The cone is cut by a plane parallel to its base soas to divide it into two parts. The volume of frustum of cone is $44{ cm }^{ 3 }$, then the radius of upper circular of frustum is $\left{ Use\ \pi =\dfrac { 22 }{ 7 } \right} $.

  1. $\sqrt [ 3 ]{ 12 } cm$
  2. $\sqrt [ 3 ]{ 13 } cm$
  3. $\sqrt [ 3 ]{ 6 } cm$
  4. $\sqrt [ 3 ]{ 20 } cm$
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B Correct answer
Explanation

Volume of cone = (1/3)pi*r^2*h = (1/3)(22/7)*9*9 = 84.85. Frustum volume = 44. Remaining small cone volume = 84.85 - 44 = 40.85. Ratio of volumes = (r_small/r_large)^3 = 40.85/84.85. Solving for r_small gives the cube root of 13.

AI explanation

The volume of the full cone is one third times pi times 3 squared times 9, which equals 84.82 cm cubed. The volume of the removed top cone is the total volume minus the frustum volume, so 84.82 minus 44 equals 40.82 cm cubed. Using the volume formula for the small cone, 40.82 equals 1/3 times 22/7 times r squared times its proportional height of 3r. Solving 40.82 equals 22/7 times r cubed gives r cubed equal to 13, meaning the radius is the cube root of 13 cm.

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