Multiple choice

A model of a building is in the from of a cone surmounting a cylinder. The radius of the cylinder is $6$ cm The height of the model is $\displaystyle 6\sqrt{3}$ cm. The vertical angle of the cone is $\displaystyle 120^{\circ}$. Find the volume of the model (in $\displaystyle cm^{2}$).

  1. $\displaystyle 148\sqrt{3}\pi $
  2. $\displaystyle 156\sqrt{3}\pi $
  3. $\displaystyle 162\sqrt{3}\pi $
  4. $\displaystyle 168\sqrt{3}\pi $
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The cone has half-angle 60°, so its height is 6/tan(60°) = 2sqrt(3) cm. The cylinder height is 6sqrt(3) - 2sqrt(3) = 4sqrt(3) cm. Adding the cylinder and cone volumes gives 168sqrt(3)π cm^3.

AI explanation

With a vertical angle of 120 degrees, the half-angle at the cone's apex is 60 degrees, and using trigonometry with the base radius of 6 cm, the cone height is 6 divided by the square root of 3, which equals 2 times the square root of 3 cm. The cylinder height is the total model height minus the cone height, so it is 6 times the square root of 3 minus 2 times the square root of 3, equaling 4 times the square root of 3 cm. Adding the cylinder volume of (pi)(6^2)(4 times the square root of 3) to the cone volume of (1/3)(pi)(6^2)(2 times the square root of 3) gives a total volume of 168 times the square root of 3 times pi.

Keep practicing — related questions