To find the total volume of the object composed of a hemisphere and a cone, we can start by denoting some variables.
Let:
- \( r \) = radius of the hemisphere
- \( h \) = height of the cone = diameter of the hemisphere = \( 2r \)
Step 1: Volume of the cone
The formula for the volume of a cone is given by: \[ V_{cone} = \frac{1}{3} \pi r^2 h \]
Substituting \( h = 2r \): \[ V_{cone} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
We know from the problem that the volume of the cone is 12 m³. Therefore, we can set up the equation: \[ \frac{2}{3} \pi r^3 = 12 \]
Now, we can solve for \( r^3 \): \[ r^3 = \frac{12 \times 3}{2 \pi} = \frac{36}{2\pi} = \frac{18}{\pi} \]
Step 2: Volume of the hemisphere
The formula for the volume of a hemisphere is: \[ V_{hemisphere} = \frac{2}{3} \pi r^3 \]
Substituting \( r^3 = \frac{18}{\pi} \) into the volume of the hemisphere: \[ V_{hemisphere} = \frac{2}{3} \pi \left(\frac{18}{\pi}\right) = \frac{2}{3} \cdot 18 = 12 \text{ m}^3 \]
Step 3: Total volume of the object
The total volume \( V_{total} \) of the object (hemisphere + cone) is given by: \[ V_{total} = V_{cone} + V_{hemisphere} = 12 \text{ m}^3 + 12 \text{ m}^3 = 24 \text{ m}^3 \]
Conclusion
The volume of the whole object is: \[ \boxed{24 \text{ m}^3} \]