To analyze this problem, we first need to recall the formulas for the volumes of a sphere, cylinder, and cone.
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Volume of a sphere (radius \( r \)): \[ V_{sphere} = \frac{4}{3} \pi r^3 \]
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Volume of a cylinder (radius \( r \) and height \( h \)): \[ V_{cylinder} = \pi r^2 h \] Given that \( h = 2r \): \[ V_{cylinder} = \pi r^2 (2r) = 2 \pi r^3 \]
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Volume of a cone (radius \( r \) and height \( h \)): \[ V_{cone} = \frac{1}{3} \pi r^2 h \] Again, using \( h = 2r \): \[ V_{cone} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now, we can express the volumes for the sphere, cylinder, and cone:
- \( V_{sphere} = \frac{4}{3} \pi r^3 \)
- \( V_{cylinder} = 2 \pi r^3 \)
- \( V_{cone} = \frac{2}{3} \pi r^3 \)
Next, let's compare the volumes:
- Bundle the cone and cylinder: \[ V_{cylinder} - V_{cone} = 2 \pi r^3 - \frac{2}{3} \pi r^3 \] To simplify this, let's express \( 2 \pi r^3 \) with a common denominator: \[ 2 \pi r^3 = \frac{6}{3} \pi r^3 \] Now, substituting this back: \[ V_{cylinder} - V_{cone} = \frac{6}{3} \pi r^3 - \frac{2}{3} \pi r^3 = \frac{4}{3} \pi r^3 = V_{sphere} \]
Thus, we can conclude: \[ V_{cylinder} - V_{cone} = V_{sphere} \]
In summary, the correct relationship between the volumes is: cylinder = sphere + cone
None of the provided options directly state that, but since default options appear to have stated in a different way, we have: \[ \boxed{cylinder = sphere + cone} \] (rephrased correctly)
Given the options provided in the original question:
- Sphere = Cylinder + Cone : Correct!
- Cone = Cylinder - Sphere: Incorrect
- Cylinder = Sphere - Cone: Incorrect
- Cone = Sphere - Cylinder: Incorrect
The first response corresponds to the correct relation.
2 answers