To analyze the relationships, let's first recall the volume formulas for a sphere, cylinder, and cone:
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Volume of a sphere (V_s): \[ V_s = \frac{4}{3} \pi r^3 \]
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Volume of a cylinder (V_c): \[ V_c = \pi r^2 h \]
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Volume of a cone (V_cone): \[ V_{cone} = \frac{1}{3} \pi r^2 h \]
For our comparison, we will assume that the height (h) is equal to the radius (r) for the cone and the cylinder. Therefore, let \( r = h \).
Substituting \( h \) with \( r \):
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For the cylinder: \[ V_c = \pi r^3 \]
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For the cone: \[ V_{cone} = \frac{1}{3} \pi r^3 \]
Now we can express the volumes in terms of \( r \):
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Volume of a sphere: \[ V_s = \frac{4}{3} \pi r^3 \]
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Volume of a cylinder: \[ V_c = \pi r^3 \]
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Volume of a cone: \[ V_{cone} = \frac{1}{3} \pi r^3 \]
Now let's derive relationships from these volumes:
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2 cylinders: \[ 2 V_c = 2 \pi r^3 \]
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3 cones: \[ 3 V_{cone} = 3 \times \frac{1}{3} \pi r^3 = \pi r^3 \]
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1 sphere: \[ V_s = \frac{4}{3} \pi r^3 \]
Now we analyze each option:
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2 cylinder = 3 cones + 1 sphere: \[ 2 \pi r^3 \overset{?}{=} \pi r^3 + \frac{4}{3} \pi r^3 \] \[ 2 \pi r^3 = \frac{7}{3} \pi r^3 \text{ (False)} \]
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2 spheres = 1 cylinder + 2 cones: \[ 2 \times \frac{4}{3} \pi r^3 \overset{?}{=} \pi r^3 + 2 \times \frac{1}{3} \pi r^3 \] \[ \frac{8}{3} \pi r^3 = \pi r^3 + \frac{2}{3} \pi r^3 = \frac{5}{3} \pi r^3 \text{ (False)} \]
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2 cylinder - 1 cone = 2 spheres: \[ 2 \pi r^3 - \frac{1}{3} \pi r^3 \overset{?}{=} 2 \times \frac{4}{3} \pi r^3 \] \[ \frac{6}{3} \pi r^3 - \frac{1}{3} \pi r^3 = \frac{12}{3} \pi r^3 \] \[ \frac{5}{3} \pi r^3 = \frac{12}{3} \pi r^3 \text{ (False)} \]
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2 spheres = 5 cones: \[ 2 \times \frac{4}{3} \pi r^3 \overset{?}{=} 5 \times \frac{1}{3} \pi r^3 \] \[ \frac{8}{3} \pi r^3 \neq \frac{5}{3} \pi r^3 \text{ (False)} \]
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2 cones = 1 cylinder: \[ 2 \times \frac{1}{3} \pi r^3 \overset{?}{=} \pi r^3 \] \[ \frac{2}{3} \pi r^3 \neq \pi r^3 \text{ (False)} \]
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2 cylinders = 3 spheres: \[ 2 \pi r^3 \overset{?}{=} 3 \times \frac{4}{3} \pi r^3 \] \[ 2 \pi r^3 \overset{?}{=} 4 \pi r^3 \text{ (False)} \]
Conclusion
None of the options provided are true. However, it's important to rely on proper calculations and relationships, as potential relationships could involve various coefficients and ratios different from those provided. But from the analysis above, we conclude that all equations presented seem incorrect.