The volume \( V \) of a cylinder is calculated using the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height. We know that the volume of Julie's sugar container is \( 288\pi \) cubic centimeters. Therefore, we can set up the equation:
\[ \pi r^2 h = 288\pi \]
Dividing both sides by \( \pi \):
\[ r^2 h = 288 \]
Now we can evaluate each of the given options.
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Radius: 3 cm, Height: 32 cm: \[ r^2 h = 3^2 \cdot 32 = 9 \cdot 32 = 288 \] Correct
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Radius: 12 cm, Height: 8 cm: \[ r^2 h = 12^2 \cdot 8 = 144 \cdot 8 = 1152 \] Not correct
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Radius: 4 cm, Height: 12 cm: \[ r^2 h = 4^2 \cdot 12 = 16 \cdot 12 = 192 \] Not correct
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Radius: 4 cm, Height: 18 cm: \[ r^2 h = 4^2 \cdot 18 = 16 \cdot 18 = 288 \] Correct
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Diameter: 12 cm, Height: 8 cm: The radius \( r = \frac{12}{2} = 6 \) cm. \[ r^2 h = 6^2 \cdot 8 = 36 \cdot 8 = 288 \] Correct
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Diameter: 16 cm, Height: 9 cm: The radius \( r = \frac{16}{2} = 8 \) cm. \[ r^2 h = 8^2 \cdot 9 = 64 \cdot 9 = 576 \] Not correct
Thus, the correct options are:
- Radius: 3 cm, Height: 32 cm
- Radius: 4 cm, Height: 18 cm
- Diameter: 12 cm, Height: 8 cm