To analyze the relationships among the volumes of the sphere, cylinder, and cone given their dimensions, we can calculate the volumes of each shape using their respective formulas.
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Volume of the Sphere (V_sphere): \[ V_{sphere} = \frac{4}{3} \pi r^3 \]
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Volume of the Cylinder (V_cylinder): The height of the cylinder (h_cylinder) is twice the radius (r), so: \[ h_{cylinder} = 2r \] The volume of the cylinder is given by: \[ V_{cylinder} = \pi r^2 h_{cylinder} = \pi r^2 (2r) = 2\pi r^3 \]
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Volume of the Cone (V_cone): The height of the cone (h_cone) is also twice the radius: \[ h_{cone} = 2r \] The volume of the cone is given by: \[ V_{cone} = \frac{1}{3} \pi r^2 h_{cone} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now we can relate these volumes:
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Sum of Cylinder and Cone: \[ V_{cylinder} + V_{cone} = 2\pi r^3 + \frac{2}{3}\pi r^3 \] To add these, we need a common denominator: \[ 2\pi r^3 = \frac{6}{3}\pi r^3 \ V_{cylinder} + V_{cone} = \frac{6}{3}\pi r^3 + \frac{2}{3}\pi r^3 = \frac{8}{3}\pi r^3 \]
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Sphere Volume: \[ V_{sphere} = \frac{4}{3} \pi r^3 \]
Next, we can compare the volumes. We can express the combined volume of the cylinder and cone in relation to the sphere:
- Compare Cylinder + Cone to Sphere: \[ V_{cylinder} + V_{cone} = \frac{8}{3}\pi r^3 \] \[ V_{sphere} = \frac{4}{3} \pi r^3 \]
We can see that: \[ V_{cylinder} + V_{cone} = \frac{8}{3} \pi r^3 \quad \text{and} \quad V_{sphere} = \frac{4}{3} \pi r^3 \]
From that, we deduce: \[ V_{cylinder} + V_{cone} = 2 \times V_{sphere} \]
Thus, none of the provided options directly represent that. However, from the options, we can see that the only one that aligns with the relationship is:
\[ \text{sphere} = \text{cylinder} - \text{cone} \]
This is incorrect, the correct expression would be; sphere = cylinder - cone is incorrect since it doesn't hold true. So, the correct option is the findings. None of the provided options strictly hold true in algebraically expressed relationships, but based on volume evaluations, we realize that both cylinder and cone volume together equal more than the sphere.
So the relationship is among those spheres but not specifically actionable in the proposals presented. Considering approximations, the suitable choice tends towards sphere equating beyond fractions contrasting cylinder and cone.
2 answers