To find the total volume of the object composed of the hemisphere and the cone, we first need to determine the volume of the hemisphere.
We know the following:
- The volume of the cone is given as \(12 , \text{m}^3\).
- The height of the cone is equal to the diameter of the hemisphere.
Let \(r\) be the radius of the hemisphere. Then the diameter of the hemisphere is \(2r\), which is also the height \(h\) of the cone.
The volume of the cone \(V_{\text{cone}}\) is given by the formula: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \] Substituting \(h = 2r\): \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \] We know that \(V_{\text{cone}} = 12 , \text{m}^3\), so: \[ \frac{2}{3} \pi r^3 = 12 \] Multiplying both sides by \(\frac{3}{2}\): \[ \pi r^3 = 18 \] Now we find \(r^3\): \[ r^3 = \frac{18}{\pi} \]
Next, we find the volume of the hemisphere \(V_{\text{hemisphere}}\): \[ V_{\text{hemisphere}} = \frac{2}{3} \pi r^3 \] Substituting \(r^3\) from above: \[ V_{\text{hemisphere}} = \frac{2}{3} \pi \left(\frac{18}{\pi}\right) = \frac{2 \cdot 18}{3} = 12 , \text{m}^3 \]
Now, we can calculate the total volume of the object: \[ V_{\text{total}} = V_{\text{cone}} + V_{\text{hemisphere}} = 12 , \text{m}^3 + 12 , \text{m}^3 = 24 , \text{m}^3 \]
The total volume of the whole object is \(24 , \text{m}^3\).
The correct answer is: 24 m³.
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