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 m3 . What is the volume of the whole object?

Question

1. 18 m3
2. 24 m3
3. There is not enough information to say.
4. 36 m3

Answer 1

To find the volume of the entire object composed of a hemisphere and a cone, we can use the following information:

1. The volume of the cone is given as \(V_{\text{cone}} = 12 , \text{m}^3\).
2. The height of the cone is equal to the diameter of the hemisphere.

Let’s break down the calculations step-by-step.

First, let’s denote:

• \( r \) = radius of the hemisphere (and thus, the radius of the cone),
• \( h \) = height of the cone.

Since the height of the cone is equal to the diameter of the hemisphere, we have: \[ h = 2r. \]

Now we can use the volume formula for the cone, which is: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h. \] Substituting \( h = 2r \) into the cone volume formula gives: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3. \]

Setting this equal to the volume of the cone we have: \[ \frac{2}{3} \pi r^3 = 12. \]

Now we can solve for \( r^3 \): \[ r^3 = \frac{12 \times 3}{2\pi} = \frac{36}{2\pi} = \frac{18}{\pi}. \]

Next, we find the volume of the hemisphere. The volume of the hemisphere is given by: \[ V_{\text{hemisphere}} = \frac{2}{3} \pi r^3. \] Substituting \( r^3 = \frac{18}{\pi} \) into the hemisphere volume formula gives: \[ V_{\text{hemisphere}} = \frac{2}{3} \pi \left( \frac{18}{\pi} \right) = \frac{2}{3} \times 18 = 12 , \text{m}^3. \]

Now, we can find the total volume of the object: \[ V_{\text{total}} = V_{\text{cone}} + V_{\text{hemisphere}} = 12 , \text{m}^3 + 12 , \text{m}^3 = 24 , \text{m}^3. \]

Thus, the volume of the whole object is: \[ \boxed{24 , \text{m}^3}. \]