An engineer wants to know the volume of the following object composed of a hemisphere and a cone. She knows the height of the cone is equal to the diameter of the hemisphere , and the volume of the cone is 12 m^3 . What is the volume of the whole object?

answers:

30 m^3

18 m^3

24 m^3

There is not enough information to say.

1 answer

To find the volume of the whole object, we need to calculate the volumes of both the hemisphere and the cone and then add them together.

  1. Volume of the Cone: The volume of the cone is given as \( V_{\text{cone}} = 12 , \text{m}^3 \).

  2. Cone Dimensions: Let's denote the radius of the base of the cone (and the hemisphere) as \( r \). The height of the cone (which is equal to the diameter of the hemisphere) is \( h = 2r \).

    The formula for the volume of a cone is: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \] Substituting \( h = 2r \): \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \] Setting this equal to the given volume of the cone: \[ \frac{2}{3} \pi r^3 = 12 \] Solving for \( r^3 \): \[ r^3 = \frac{12 \times 3}{2\pi} = \frac{36}{2\pi} = \frac{18}{\pi} \]

  3. Volume of the Hemisphere: The volume of a hemisphere is given by: \[ V_{\text{hemisphere}} = \frac{2}{3} \pi r^3 \] Now we can substitute \( r^3 = \frac{18}{\pi} \): \[ V_{\text{hemisphere}} = \frac{2}{3} \pi \left(\frac{18}{\pi}\right) = \frac{2 \times 18}{3} = 12 \]

  4. Total Volume: The total volume of the whole object (the hemisphere plus the cone) is: \[ V_{\text{total}} = V_{\text{cone}} + V_{\text{hemisphere}} = 12 + 12 = 24 , \text{m}^3 \]

Thus, the volume of the whole object is \( 24 , \text{m}^3 \).

The correct answer is: 24 m³.