To find the volume of the whole 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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Given Information:
- Volume of the cone = 12 m³
- Height of the cone = diameter of the hemisphere.
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Relationship Between Cone and Hemisphere: Since the height of the cone is equal to the diameter of the hemisphere, if we let the radius of the hemisphere be \( r \), then:
- Diameter of the hemisphere = \( 2r \)
- Height of the cone = \( 2r \)
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Volume of the Hemisphere: The volume \( V \) of a hemisphere is given by the formula: \[ V = \frac{2}{3} \pi r^3 \]
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Volume of the Cone: The volume \( V \) of a cone is given by the formula: \[ V = \frac{1}{3} \pi r^2 h \] Substituting \( h = 2r \): \[ V = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
However, we know that the volume of the cone is given as 12 m³: \[ \frac{2}{3} \pi r^3 = 12 \]
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Finding the Volume of the Hemisphere: From the above equation, we can find the total volume of the whole object:
- We can calculate the volume of the hemisphere:
Since we already find the volume of the cone as \( \frac{2}{3} \pi r^3 \), which equals 12 m³, it implies that for the hemisphere: \[ V_{\text{hemisphere}} = 12 , \text{m}^3 \]
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Total Volume: Thus, the total volume of the whole object: \[ V_{\text{total}} = V_{\text{cone}} + V_{\text{hemisphere}} = 12 , \text{m}^3 + 12 , \text{m}^3 = 24 , \text{m}^3 \]
So, the total volume of the whole object is 24 m³.
Final Answer: 24 m³