To determine the correct relationship between the volumes of the sphere, cylinder, and cone, we first need to calculate their 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 (h) of the cylinder is twice the radius (r), so \( h = 2r \). \[ V_c = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3 \]
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Volume of the Cone (V_cone): The height (h) of the cone is also \( h = 2r \). \[ V_{cone} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now we can compare the volumes:
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Volume of the Sphere: \[ V_s = \frac{4}{3} \pi r^3 \]
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Volume of the Cylinder: \[ V_c = 2\pi r^3 \]
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Volume of the Cone: \[ V_{cone} = \frac{2}{3} \pi r^3 \]
Next, we can rewrite \(V_{c}\) and \(V_{cone}\) to express them in terms of \(V_s\):
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Expressing the Cylinder in terms of the Sphere and Cone: \[ V_c = V_s - V_{cone} \]
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Substituting volumes:
- \(V_s = \frac{4}{3} \pi r^3\)
- \(V_{cone} = \frac{2}{3} \pi r^3\)
So: \[ V_c = \frac{4}{3} \pi r^3 - \frac{2}{3} \pi r^3 = \frac{2}{3} \pi r^3 \] This means the formula \(V_c = V_s - V_{cone}\) holds true.
Now let's examine each response option:
- cylinder = sphere – cone: (This is true)
- cone = sphere – cylinder: (This is false)
- sphere = cylinder + cone: (This is false)
- cone = cylinder – sphere: (This is false)
Thus, the correct relationship is:
cylinder = sphere – cone