To calculate the volume of a regular hexagonal prism, we can follow these steps:
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Calculate the area of the hexagonal base: The area \( A \) of a regular hexagon can be calculated using the formula:
\[ A = \frac{3 \sqrt{3}}{2} s^2 \]
where \( s \) is the length of a side of the hexagon.
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Calculate the volume of the prism: The volume \( V \) of the prism is then calculated as:
\[ V = A \cdot h \]
where \( h \) is the height of the prism.
Given Dimensions
- Side length \( s = 8 \) cm
- Height \( h = 22 \) cm
Calculation Steps
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Calculate the area of the hexagon: \[ A = \frac{3 \sqrt{3}}{2} (8^2) = \frac{3 \sqrt{3}}{2} (64) = 96\sqrt{3} \]
Using \( \sqrt{3} \approx 1.732\): \[ A \approx 96 \times 1.732 \approx 165.12 , \text{cm}^2 \]
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Calculate the volume: \[ V = A \cdot h = 165.12 \cdot 22 \approx 3632.64 , \text{cm}^3 \]
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Round to the nearest cubic centimeter: \[ V \approx 3633 , \text{cm}^3 \]
Thus, the correct volume of the regular hexagonal prism is approximately 3633 cm³.
Analyzing the Error
Looking at the error options:
- A. The student multiplied 2 times the three given measurements. - This is not logical for volume.
- B. The student multiplied the perimeter of the base times the height of the prism. - This would give a linear volume but isn’t standard formula.
- C. The student multiplied the three given measurements. - This is not specific enough about the method.
- D. The student multiplied the height of the prism times the square of the side length of the hexagon. - This would yield a wrong understanding since it doesn't account for the area of the base.
The most likely error made by the student was:
D. The student multiplied the height of the prism times the square of the side length of the hexagon.
This would suggest they were misunderstanding the area of the hexagon and erroneously calculating a volume without considering the proper area of the base.