To find the volume of the whole object composed of a hemisphere and a cone, we'll start with the volume of the cone and then add the volume of the hemisphere.
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Volume of the Cone: The volume of the cone is given as 12 m³.
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Volume of the Hemisphere: The formula for the volume of a hemisphere is given by: \[ V = \frac{2}{3} \pi r^3 \] Since the height of the cone is equal to the diameter of the hemisphere, this means that:
- If \( r \) is the radius of the hemisphere, then the diameter is \( 2r \).
- Therefore, the height of the cone \( h = 2r \).
The volume of the cone can also be expressed as: \[ V_{cone} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \] We know from the problem statement that \( V_{cone} = 12 , m^3 \), therefore: \[ \frac{2}{3} \pi r^3 = 12 \]
To find \( r^3 \): \[ \pi r^3 = 12 \times \frac{3}{2} = 18 \implies r^3 = \frac{18}{\pi} \]
Now substituting \( r^3 \) back into the volume formula for the hemisphere: \[ V_{hemisphere} = \frac{2}{3} \pi r^3 = \frac{2}{3} \pi \left( \frac{18}{\pi} \right) = 12 \]
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Total Volume: Now we can find the total volume of the object: \[ V_{total} = V_{cone} + V_{hemisphere} = 12 , m^3 + 12 , m^3 = 24 , m^3 \]
Thus, the volume of the whole object is 24 m³.