To calculate the volumes of the three figures, we can use the following formulas:
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Volume of a Cone: \[ V = \frac{1}{3} \pi r^2 h \] where \( r \) is the radius, and \( h \) is the height.
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Volume of a Cube: \[ V = a^3 \] where \( a \) is the length of each side.
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Volume of a Rectangular Prism: \[ V = l \times w \times h \] where \( l \) is the length, \( w \) is the width, and \( h \) is the height.
Now, let's calculate the volumes step-by-step.
Volume of the Cone
- Radius \( r = 5 \)
- Height \( h = 8.25 \)
\[ V = \frac{1}{3} \pi (5)^2 (8.25) = \frac{1}{3} \pi (25)(8.25) = \frac{1}{3} \pi (206.25) \approx \frac{206.25}{3} \pi \approx 68.75 \pi \approx 216.00 \text{ (using } \pi \approx 3.14 \text{)} \]
Volume of the Cube
- Length \( a = 6 \)
\[ V = 6^3 = 216 \]
Volume of the Rectangular Prism
- Length \( l = 6 \), Width \( w = 3 \), Height \( h = 12 \)
\[ V = 6 \times 3 \times 12 = 216 \]
Summary of the Volumes
- Cone Volume: \( \approx 216.00 \)
- Cube Volume: \( 216 \)
- Rectangular Prism Volume: \( 216 \)
Analysis of Findings
When we calculated the volumes of the cone, cube, and rectangular prism, we found that the volumes of the cube and rectangular prism were exactly the same, while the volume of the cone was approximately equal to these values.
The reason for these results is that the cube and the rectangular prism have a volume of \( 216 \) cubic units because they are both solid shapes with regular geometric properties defined by simple multiplication of their dimensions. On the other hand, the cone has a volume that is a third of the volume of a cylinder and takes into account the shape tapering to a point, which reduces its overall volume compared to the rectangular shapes.
In conclusion, the cone's volume approximates that of the cube and rectangular prism, thus highlighting the differences in measurements among shapes. The similarity in volume suggests that while the total occupied space might be different in shape, in this case, they contain about the same amount of volume.