To find the volume of the entire object, which consists of a hemisphere placed on top of a cone, we first need to calculate the volume of the hemisphere and then add it to the volume of the cone.
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Volume of the Cone: The volume of the cone is given as \( V_{cone} = 12 , m^3 \).
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Finding the radius of the cone and hemisphere: We know that the height of the cone is equal to the diameter of the hemisphere. Let the radius of the hemisphere be \( r \). Therefore, the diameter of the hemisphere is \( 2r \), meaning the height of the cone \( h_{cone} = 2r \).
The volume of a cone is given by the formula: \[ V_{cone} = \frac{1}{3} \pi r^2 h_{cone} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Set this equal to the given volume of the cone: \[ \frac{2}{3} \pi r^3 = 12 \]
Solve for \( r^3 \): \[ r^3 = \frac{12 \cdot 3}{2\pi} = \frac{36}{2\pi} = \frac{18}{\pi} \]
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Volume of the Hemisphere: The volume of a hemisphere is given by the formula: \[ V_{hemisphere} = \frac{2}{3} \pi r^3 \]
Substituting \( r^3 \): \[ V_{hemisphere} = \frac{2}{3} \pi \left(\frac{18}{\pi}\right) = \frac{2 \cdot 18}{3} = 12 \]
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Total Volume of the Object: Now, we add the volumes of the cone and hemisphere to find the total volume: \[ V_{total} = V_{cone} + V_{hemisphere} = 12 , m^3 + 12 , m^3 = 24 , m^3 \]
Thus, the volume of the whole object is \( \boxed{24 , m^3} \).