To find the correct relationship between the volumes of the sphere, cylinder, and cone, we need to calculate the volume of each shape based on the given conditions.
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Sphere: The volume \( V_s \) of a sphere with radius \( r \) is given by: \[ V_s = \frac{4}{3} \pi r^3 \]
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Cylinder: The volume \( V_c \) of a cylinder with radius \( r \) and height \( h \) is given by: \[ V_c = \pi r^2 h \] Given that the height \( h \) is twice the radius \( r \), we have \( h = 2r \): \[ V_c = \pi r^2 (2r) = 2\pi r^3 \]
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Cone: The volume \( V_{cone} \) of a cone with radius \( r \) and height \( h \) is given by: \[ V_{cone} = \frac{1}{3} \pi r^2 h \] Again, since the height \( h \) is \( 2r \): \[ V_{cone} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now we have:
- Volume of the sphere: \( V_s = \frac{4}{3} \pi r^3 \)
- Volume of the cylinder: \( V_c = 2\pi r^3 \)
- Volume of the cone: \( V_{cone} = \frac{2}{3} \pi r^3 \)
Next, let’s evaluate the relationships given:
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Check: \( 1 \text{ cylinder} = 2 \text{ cones} + 1 \text{ sphere} \) \[ V_c = 2V_{cone} + V_s \] Substitute: \[ 2 \left(\frac{2}{3} \pi r^3\right) + \frac{4}{3} \pi r^3 = \frac{4}{3} \pi r^3 + \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3 , (\text{which is } \neq 2 \pi r^3) \]
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Check: \( 1 \text{ cylinder} = 2 \text{ cones} + 1 \text{ sphere} \)
- NOT true (from above check).
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Check: \( \text{sphere} - \text{cone} = \text{cylinder} \) \[ V_s - V_{cone} = \frac{4}{3} \pi r^3 - \frac{2}{3} \pi r^3 = \frac{2}{3} \pi r^3 , (\text{meaning this } \text{is NOT equal to } 2 \pi r^3) \]
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Check: \( \text{sphere} - \text{cone} = \text{cylinder} \) - NOT true (as calculated).
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Check: \( 3 \text{ cones} = 1 \text{ cylinder} \) \[ 3V_{cone} = 3 \left(\frac{2}{3} \pi r^3\right) = 2 \pi r^3 = V_c , (\text{This is true}) \]
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Check: \( 1 \text{ cylinder} = 3 \text{ cones} \) - True.
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Check: \( \text{sphere} = \text{cylinder} + \text{cone} \) \[ V_s = V_c + V_{cone} = 2\pi r^3 + \frac{2}{3} \pi r^3 = \frac{6}{3} \pi r^3 + \frac{2}{3} \pi r^3 = \frac{8}{3} \pi r^3 , (\text{shown false}) \]
Based on the calculations, the correct relationship is: \[ \textbf{3 cones} = 1 \text{ cylinder} \]