To find the total volume of the object composed of a hemisphere and a cone, we need to first find the volume of the hemisphere given the information provided.
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Volume of the cone (Vc): The volume of the cone is given as 12 m³.
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Height of the cone (h): The height of the cone is equal to the diameter of the hemisphere. Let \( r \) be the radius of the hemisphere. Therefore, the diameter \( d \) of the hemisphere is \( 2r \), which means the height of the cone is also \( 2r \).
The volume of a cone is given by the formula: \[ V_c = \frac{1}{3} \pi r^2 h \] Substituting \( h = 2r \): \[ V_c = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \] Setting this equal to the given volume of the cone: \[ \frac{2}{3} \pi r^3 = 12 \]
To find \( r^3 \): \[ r^3 = 12 \cdot \frac{3}{2\pi} = \frac{36}{2\pi} = \frac{18}{\pi} \]
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Volume of the hemisphere (Vh): The volume of a hemisphere is given by the formula: \[ V_h = \frac{2}{3} \pi r^3 \] Substituting \( r^3 \): \[ V_h = \frac{2}{3} \pi \left(\frac{18}{\pi}\right) = \frac{2 \cdot 18}{3} = 12 m^3 \]
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Total volume (V_total): The total volume of the object composed of both the cone and the hemisphere is: \[ V_{total} = V_c + V_h = 12 m^3 + 12 m^3 = 24 m^3 \]
Thus, the volume of the whole object is 24 m³.