To find the total volume of the object composed of a hemisphere and a cone, we will use the known volumes and relationships.
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Volume of the cone: The volume of the cone is given as \(12 , \text{m}^3\).
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Cone's height and hemisphere's dimensions: It is stated 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 means the height of the cone \(h\) is also \(2r\).
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We can use the formula for the volume of a cone:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
Substituting \(h = 2r\):
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
We know that the volume of the cone is \(12 , \text{m}^3\):
\[ \frac{2}{3} \pi r^3 = 12 \]
This can be rearranged to find \(r^3\):
\[ r^3 = \frac{12 \times 3}{2 \pi} = \frac{36}{2 \pi} = \frac{18}{\pi} \]
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Volume of the hemisphere: The volume of a hemisphere is given by the formula:
\[ V_{\text{hemisphere}} = \frac{2}{3} \pi r^3 \]
Substituting in our expression for \(r^3\):
\[ V_{\text{hemisphere}} = \frac{2}{3} \pi \left(\frac{18}{\pi}\right) = \frac{2 \times 18}{3} = 12 , \text{m}^3 \]
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Total volume: The total volume of the object is the sum of 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 total volume of the whole object is 24 m³.
The correct response is: 24 m cubed.
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