To find the volume of the object composed of a hemisphere and a cone, we 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: We're given that the volume of the cone is \(12 , m^3\).
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Volume of the Hemisphere: The volume \(V\) of a hemisphere is given by the formula: \[ V = \frac{2}{3} \pi r^3 \] Where \(r\) is the radius of the hemisphere.
Since the height of the cone is equal to the diameter of the hemisphere, we can express the diameter \(d\) of the hemisphere as: \[ d = 2r \] Therefore, the height \(h\) of the cone is: \[ h = 2r \]
The volume of a cone is given by: \[ V = \frac{1}{3} \pi r^2 h \] Substituting \(h = 2r\) into the volume formula for the cone gives: \[ V_{cone} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
According to the problem, we know the volume of the cone is \(12 , m^3\): \[ \frac{2}{3} \pi r^3 = 12 \] To find \(r^3\), we can solve for it: \[ r^3 = \frac{12 \cdot 3}{2\pi} = \frac{36}{2\pi} = \frac{18}{\pi} \]
Now, we can calculate the volume of the hemisphere: \[ V_{hemisphere} = \frac{2}{3} \pi r^3 = \frac{2}{3} \pi \left(\frac{18}{\pi}\right) = \frac{2}{3} \cdot 18 = 12 , m^3 \]
- Total Volume: Now we add the volumes of the cone and the hemisphere: \[ V_{total} = V_{cone} + V_{hemisphere} = 12 , m^3 + 12 , m^3 = 24 , m^3 \]
Thus, the volume of the whole object is:
\[ \boxed{24 , m^3} \]
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