To find the volume of the whole object, we need to calculate the volumes of both the hemisphere and the cone and then add them together.
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Volume of the Cone: The volume of the cone is given as \( V_{\text{cone}} = 12 , \text{m}^3 \).
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Cone Dimensions: Let's denote the radius of the base of the cone (and the hemisphere) as \( r \). The height of the cone (which is equal to the diameter of the hemisphere) is \( h = 2r \).
The formula for the volume of a cone is: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \] Substituting \( h = 2r \): \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \] Setting this equal to the given volume of the cone: \[ \frac{2}{3} \pi r^3 = 12 \] Solving for \( r^3 \): \[ r^3 = \frac{12 \times 3}{2\pi} = \frac{36}{2\pi} = \frac{18}{\pi} \]
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Volume of the Hemisphere: The volume of a hemisphere is given by: \[ V_{\text{hemisphere}} = \frac{2}{3} \pi r^3 \] Now we can substitute \( r^3 = \frac{18}{\pi} \): \[ V_{\text{hemisphere}} = \frac{2}{3} \pi \left(\frac{18}{\pi}\right) = \frac{2 \times 18}{3} = 12 \]
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Total Volume: The total volume of the whole object (the hemisphere plus the cone) is: \[ V_{\text{total}} = V_{\text{cone}} + V_{\text{hemisphere}} = 12 + 12 = 24 , \text{m}^3 \]
Thus, the volume of the whole object is \( 24 , \text{m}^3 \).
The correct answer is: 24 m³.