To find the total volume of the object composed of a hemisphere and a cone, we need to determine the volume of the hemisphere and add it to the volume of the cone.
We know the following:
- The volume of the cone (V_cone) is given as 12 m³.
- The height of the cone (h_cone) is equal to the diameter of the hemisphere (d_hemisphere).
Let's denote the radius of the hemisphere as \( r \). Then:
- The diameter of the hemisphere is \( d_{hemisphere} = 2r \).
- The height of the cone is also \( h_{cone} = 2r \).
The volume of a cone is given by the formula: \[ V_{cone} = \frac{1}{3} \pi r^2 h \] Substituting \( h \) with \( 2r \): \[ V_{cone} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \] We set this equal to the given volume of the cone: \[ \frac{2}{3} \pi r^3 = 12 \] Solving for \( r^3 \): \[ r^3 = \frac{12 \cdot 3}{2 \pi} = \frac{36}{2 \pi} = \frac{18}{\pi} \]
Next, we determine the volume of the hemisphere using the formula: \[ V_{hemisphere} = \frac{2}{3} \pi r^3 \] Substituting \( r^3 \) from above: \[ V_{hemisphere} = \frac{2}{3} \pi \left(\frac{18}{\pi}\right) = \frac{2 \cdot 18}{3} = \frac{36}{3} = 12 \text{ m}^3 \]
Now, we can find the total volume of the object: \[ V_{total} = V_{cone} + V_{hemisphere} = 12 \text{ m}^3 + 12 \text{ m}^3 = 24 \text{ m}^3 \]
Thus, the volume of the whole object is:
24 m³