To analyze the relationship between the volumes of a sphere, cylinder, and cone with the same radius (r), and the cylinder and cone having a height (h) that is twice the radius, we start by calculating the volumes using their respective formulas.
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Volume of the Sphere (V_s): \[ V_s = \frac{4}{3} \pi r^3 \]
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Volume of the Cylinder (V_c): The height of the cylinder is \(h = 2r\). \[ V_c = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3 \]
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Volume of the Cone (V_n): The height of the cone is also \(h = 2r\). \[ V_n = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now we can compare the volumes:
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Relationship between 2 spheres and the cone: \[ 2V_s = 2 \left(\frac{4}{3} \pi r^3\right) = \frac{8}{3} \pi r^3 \quad \text{and} \quad V_n = \frac{2}{3} \pi r^3 \] So, \(2V_s \neq V_n\).
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Relationship between 2 cylinders and the cone: \[ 2V_c = 2(2\pi r^3) = 4\pi r^3 \quad \text{and} \quad V_n = \frac{2}{3} \pi r^3 \] So, \(2V_c \neq V_n\).
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Relationship between 2 cones and the sphere: \[ 2V_n = 2\left(\frac{2}{3} \pi r^3\right) = \frac{4}{3} \pi r^3 \quad \text{and} \quad V_s = \frac{4}{3} \pi r^3 \] So, \(2V_n = V_s\).
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Relationship between 2 cones and the cylinder: \[ 2V_n = 2 \left(\frac{2}{3} \pi r^3\right) = \frac{4}{3} \pi r^3 \quad \text{and} \quad V_c = 2\pi r^3 \] So, \(2V_n \neq V_c\).
Now we can conclude:
The only true statement is: 2 cones = 1 sphere.