To find the volume of the object composed of a hemisphere and a cone, we first need to use the information provided about the cone and its relation to the hemisphere.
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Volume of the Cone: We know that the volume of the cone is given as \( V_{\text{cone}} = 12 , \text{m}^3 \).
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Relation between the Cone and Hemisphere: The problem states that the height of the cone is equal to the diameter of the hemisphere. Let's denote:
- The radius of the hemisphere as \( r \).
- Therefore, the diameter of the hemisphere is \( 2r \), which is also the height of the cone.
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Volume Formula for the Cone: The volume \( V \) of a cone can be calculated using the formula: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \] In this case, we have:
- \( h = 2r \) (height of the cone) Substituting this into the volume formula gives: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \] We know that \( V_{\text{cone}} = 12 , \text{m}^3 \). Therefore, we equate: \[ \frac{2}{3} \pi r^3 = 12 \] Multiplying both sides by \( \frac{3}{2} \) yields: \[ \pi r^3 = 18 \] Dividing both sides by \( \pi \): \[ r^3 = \frac{18}{\pi} \]
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Volume of the Hemisphere: The volume \( V \) of a hemisphere is given by the formula: \[ V_{\text{hemisphere}} = \frac{2}{3} \pi r^3 \] Using the value of \( r^3 \) we found, we can substitute to find the volume of the hemisphere: \[ V_{\text{hemisphere}} = \frac{2}{3} \pi \left(\frac{18}{\pi}\right) \] Simplifying this: \[ V_{\text{hemisphere}} = \frac{2}{3} \cdot 18 = 12 , \text{m}^3 \]
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Total Volume of the Object: Finally, we add the volumes of the cone and the hemisphere: \[ V_{\text{total}} = V_{\text{cone}} + V_{\text{hemisphere}} = 12 , \text{m}^3 + 12 , \text{m}^3 = 24 , \text{m}^3 \]
Thus, the volume of the whole object is 24 m³.