To analyze the relationships between the volumes of a sphere, cylinder, and cone, we can start with the volume formulas for each shape:
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Volume of the sphere (V_s): \[ V_s = \frac{4}{3} \pi r^3 \] where \( r \) is the radius.
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Volume of the cylinder (V_c): \[ V_c = \pi r^2 h \] where \( h \) is the height and \( r \) is the radius.
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Volume of the cone (V_{cone}): \[ V_{cone} = \frac{1}{3} \pi r^2 h \]
If we assume that the cone and cylinder have the same height \( h \) and share the same radius \( r \):
Now let's evaluate the statements:
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sphere = cylinder + cone \[ \frac{4}{3} \pi r^3 \neq \pi r^2 h + \frac{1}{3} \pi r^2 h \] (Not true, as the left-hand side scales with \( r^3 \) while the right scales with \( r^2 \) and \( h \)).
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3 cones = 1 cylinder \[ 3 \left( \frac{1}{3} \pi r^2 h \right) = \pi r^2 h \] (True: The volumes are equal).
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1 cylinder = 2 cones + 1 sphere \[ \pi r^2 h \neq 2 \left( \frac{1}{3} \pi r^2 h \right) + \frac{4}{3} \pi r^3 \] (Not true, as the dimensions are inconsistent).
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sphere - cone = cylinder \[ \frac{4}{3} \pi r^3 - \left( \frac{1}{3} \pi r^2 h \right) \neq \pi r^2 h \] (Not true, again inconsistencies in terms).
Conclusion:
The only true statement from the options presented is:
2. 3 cones = 1 cylinder.
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