To analyze the volume relationships among a sphere, cylinder, and cone with the same radius \( r \) and the cylinder and cone sharing the same height \( h \), we can use their volume formulas:
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Volume of a sphere: \[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
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Volume of a cylinder: \[ V_{\text{cylinder}} = \pi r^2 h \]
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Volume of a cone: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
Now, let's evaluate the proposed relationships one by one:
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sphere = cylinder + cone: \[ \frac{4}{3} \pi r^3 \neq \pi r^2 h + \frac{1}{3} \pi r^2 h \]
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3 cones = 1 cylinder: \[ 3 \left( \frac{1}{3} \pi r^2 h \right) = \pi r^2 h \] This is true since \( 3 \) cones equal \( 1 \) cylinder.
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1 cylinder = 2 cones + 1 sphere: \[ \pi r^2 h \neq 2 \left( \frac{1}{3} \pi r^2 h \right) + \frac{4}{3} \pi r^3 \]
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sphere - cone = cylinder: \[ \frac{4}{3} \pi r^3 - \frac{1}{3} \pi r^2 h \neq \pi r^2 h \]
From this analysis, only the relationship:
3 cones = 1 cylinder
is true. Therefore, the correct answer is:
3 cones = 1 cylinder.