To find the volume of the entire object composed of a hemisphere and a cone, we can follow these steps:
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Understand the relationship between the cone and the hemisphere: The problem states that the height of the cone is equal to the diameter of the hemisphere.
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Express the volumes:
- The volume of the cone \( V_{cone} \) is given as 12 m³.
- The volume of a hemisphere \( V_{hemisphere} \) can be found using the formula: \[ V_{hemisphere} = \frac{2}{3} \pi r^3 \] where \( r \) is the radius of the hemisphere.
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Relate the cone and hemisphere dimensions: Since the height of the cone equals the diameter of the hemisphere, we can write:
- Height of the cone \( h_{cone} = 2r \).
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Find the volume of the cone using its formula: \[ V_{cone} = \frac{1}{3} \pi r^2 h_{cone} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
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Set the volume of the cone equal to its given value: \[ \frac{2}{3} \pi r^3 = 12 \text{ m}^3 \]
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Solve for \( r^3 \): \[ r^3 = \frac{12 \times 3}{2 \pi} = \frac{36}{2 \pi} = \frac{18}{\pi} \]
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Calculate the volume of the hemisphere: \[ V_{hemisphere} = \frac{2}{3} \pi r^3 = \frac{2}{3} \pi \left(\frac{18}{\pi}\right) = \frac{2}{3} \times 18 = 12 \text{ m}^3 \]
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Total volume of the object: \[ V_{total} = V_{cone} + V_{hemisphere} = 12 \text{ m}^3 + 12 \text{ m}^3 = 24 \text{ m}^3 \]
Thus, the volume of the whole object is 24 m³.