To find the volume of the entire object, we first need to determine the volumes of both the hemisphere and the cone.
We know that:
- The volume of the cone \( V_{cone} = 12 , m^3 \).
- The height of the cone is equal to the diameter of the hemisphere.
Step 1: Volume of the Cone
The volume of a cone is given by the formula: \[ V_{cone} = \frac{1}{3} \pi r^2 h \] where \( r \) is the radius of the cone's base and \( h \) is the height of the cone.
Given that \( V_{cone} = 12 , m^3 \), we can write: \[ 12 = \frac{1}{3} \pi r_{cone}^2 h_{cone} \]
Step 2: Relationship Between Cone and Hemisphere
Let the radius of the hemisphere be \( r_{hemisphere} \). The height of the cone \( h_{cone} \) is equal to the diameter of the hemisphere, which is \( 2r_{hemisphere} \). Thus: \[ h_{cone} = 2r_{hemisphere} \]
From this, we can also derive that: \[ h_{cone} = 2r_{cone} \] where \( r_{cone} = r_{hemisphere} \) (since they both share the same base).
Step 3: Substitute into the Cone Volume Formula
Now, using the above relationship, we can substitute into the cone's volume formula: \[ 12 = \frac{1}{3} \pi r_{hemisphere}^2 (2r_{hemisphere}) \] Simplifying this, we get: \[ 12 = \frac{2}{3} \pi r_{hemisphere}^3 \]
Step 4: Solve for Radius of Hemisphere
Now we solve for \( r_{hemisphere} \): \[ 12 = \frac{2}{3} \pi r_{hemisphere}^3 \] \[ r_{hemisphere}^3 = \frac{12 \cdot 3}{2 \pi} = \frac{36}{2\pi} = \frac{18}{\pi} \] \[ r_{hemisphere} = \left(\frac{18}{\pi}\right)^{\frac{1}{3}} \]
Step 5: Volume of the Hemisphere
The volume of a hemisphere is given by: \[ V_{hemisphere} = \frac{2}{3} \pi r_{hemisphere}^3 \] Substituting \( r_{hemisphere}^3 \): \[ V_{hemisphere} = \frac{2}{3} \pi \left(\frac{18}{\pi}\right) \] \[ V_{hemisphere} = \frac{2}{3} \cdot 18 = 12 , m^3 \]
Step 6: Total Volume
Now we can find the total volume of the whole object: \[ V_{total} = V_{hemisphere} + V_{cone} = 12 , m^3 + 12 , m^3 = 24 , m^3 \]
Thus, the volume of the entire object is: \[ \boxed{24 , m^3} \]
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