To find the volume of the whole object composed of a hemisphere and a cone, we can follow these steps:
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Volume of the Cone: You mentioned that the volume of the cone is 12 m³.
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Relationship between the Cone and Hemisphere: 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 \( 2r \), and the height of the cone is \( 2r \).
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Volume of the Hemisphere: The formula for the volume of a hemisphere is given by: \[ V_{\text{hemisphere}} = \frac{2}{3} \pi r^3 \]
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Finding \( r \): We need to relate the radius \( r \) of the hemisphere to the volume of the cone. The volume of the cone is calculated using the formula: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \] Given that the height \( h \) of the cone is \( 2r \), we can express the volume of the cone as: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
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Given the Volume of the Cone: We know that \( V_{\text{cone}} = 12 , \text{m}^3 \): \[ \frac{2}{3} \pi r^3 = 12 \]
To find \( r^3 \): \[ r^3 = 12 \times \frac{3}{2 \pi} = \frac{36}{2 \pi} = \frac{18}{\pi} \]
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Calculating the Volume of the Hemisphere: Now we can calculate the volume of the hemisphere: \[ V_{\text{hemisphere}} = \frac{2}{3} \pi r^3 = \frac{2}{3} \pi \left( \frac{18}{\pi} \right) = \frac{36}{3} = 12 , \text{m}^3 \]
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Total Volume of the Object: Finally, the total volume of the object (hemisphere + cone) is: \[ V_{\text{total}} = V_{\text{hemisphere}} + V_{\text{cone}} = 12 , \text{m}^3 + 12 , \text{m}^3 = 24 , \text{m}^3 \]
So, the correct response is: 24 m³ or 24 m cubed.