Bot GPT 4-o mini To find the total volume of the object, we need to calculate both the volume of the cone and the volume of the hemisphere, and then add them together.
We are given:
- The volume of the cone \( V_{cone} = 12 , m^3 \).
- The height of the cone is equal to the diameter of the hemisphere.
Let’s denote:
- The radius of the hemisphere as \( r \).
- The diameter of the hemisphere (and the height of the cone) as \( d = 2r \).
The volume of the hemisphere can be calculated using the formula: \[ V_{hemisphere} = \frac{2}{3} \pi r^3 \]
Now, the volume of the cone can be calculated using the formula: \[ V_{cone} = \frac{1}{3} \pi r^2 h \] Since the height \( h \) of the cone is equal to the diameter of the hemisphere, we have: \[ h = d = 2r \] Substituting this back into the cone's volume formula gives us: \[ V_{cone} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Since we know the volume of the cone is 12 m³, we can set up the equation: \[ \frac{2}{3} \pi r^3 = 12 \] Solving for \( r^3 \): \[ r^3 = 12 \cdot \frac{3}{2\pi} = \frac{36}{2\pi} = \frac{18}{\pi} \]
Next, we can calculate the volume of the hemisphere: \[ V_{hemisphere} = \frac{2}{3} \pi r^3 = \frac{2}{3} \pi \left( \frac{18}{\pi} \right) = \frac{2}{3} \cdot 18 = 12 , m^3 \]
Now, we can find the total volume of the object: \[ V_{total} = V_{cone} + V_{hemisphere} = 12 , m^3 + 12 , m^3 = 24 , m^3 \]
The volume of the whole object is \( 24 , m^3 \).
Therefore, the answer is: 24 m³ or 24 m cubed.
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