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: \[ V = s^3 \] where \( s \) is the length of a side of the cube.
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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 of each figure:
1. Volume of the Cone:
- Given: radius \( r = 5 \), height \( h = 8.25 \) \[ V_\text{cone} = \frac{1}{3} \pi (5)^2 (8.25) = \frac{1}{3} \pi (25)(8.25) = \frac{1}{3} \pi (206.25) \approx 216.99 \text{ cubic units} \]
2. Volume of the Cube:
- Given: length \( s = 6 \) \[ V_\text{cube} = 6^3 = 216 \text{ cubic units} \]
3. Volume of the Rectangular Prism:
- Given: length \( l = 6 \), width \( w = 3 \), height \( h = 12 \) \[ V_\text{prism} = 6 \times 3 \times 12 = 216 \text{ cubic units} \]
Summary of Volumes:
- Volume of Cone: \( \approx 216.99 \text{ cubic units} \)
- Volume of Cube: \( 216 \text{ cubic units} \)
- Volume of Rectangular Prism: \( 216 \text{ cubic units} \)
Findings:
When calculating the volumes, we find that the cone's volume is approximately equal to the volumes of the cube and the rectangular prism, which are identically 216 cubic units. This occurs because the cone's dimensions (radius and height) were chosen such that its volume is very close to those of the other two shapes.
Explanation:
The slight difference in volume for the cone is due to the nature of its geometry; it has a conical shape which takes up less space than a cube or a rectangular prism of the same height and width when viewed in three-dimensional space. The mathematical relationship also shows that while the cone appears to occupy a similar volume, it is still less due to it tapering to a single point, which reduces the total volume compared to solid figures with flat faces like the cube or prism.