To determine the volume of the object composed of a hemisphere and a cone, we will first need to establish the relationship between the dimensions given.
-
Volume of the Cone: We know the volume of the cone is \( V_{cone} = 12 , m^3 \).
-
Relationship between the cone and the hemisphere:
- The height of the cone (h) is equal to the diameter of the hemisphere (d), which is twice the radius (r) of the hemisphere.
- Therefore, we have:
- \( h = d = 2r \)
-
Volume of the Hemisphere:
- The volume of a hemisphere is given by the formula: \[ V_{hemisphere} = \frac{2}{3} \pi r^3 \]
-
Volume of the Cone Formula:
- The volume of a cone is given by the formula: \[ V_{cone} = \frac{1}{3} \pi r^2 h \]
- Since \( h = 2r \), we can replace \( h \) in the cone's volume formula: \[ V_{cone} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
-
Equating Volumes:
- We know from the problem statement that \( V_{cone} = 12 , m^3 \): \[ \frac{2}{3} \pi r^3 = 12 \]
- Solving for \( r^3 \): \[ r^3 = \frac{12 \cdot 3}{2\pi} = \frac{36}{2\pi} = \frac{18}{\pi} \]
-
Calculating Volume of the Hemisphere:
- Plugging \( r^3 \) back into the hemisphere volume formula: \[ V_{hemisphere} = \frac{2}{3} \pi r^3 = \frac{2}{3} \pi \left(\frac{18}{\pi}\right) = \frac{2 \cdot 18}{3} = 12 \]
-
Total Volume of the Object:
- Now, we can find the total volume of the object (hemisphere + cone): \[ V_{total} = V_{hemisphere} + V_{cone} = 12 , m^3 + 12 , m^3 = 24 , m^3 \]
Thus, the volume of the whole object is 24 m³.