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in Surface Areas And Volumes by (46.2k points)
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There are two identical cubes. Out of one cube, a sphere of maximum volume (VS) is cut off. Out of the second cube, a cone of maximum volume (VC) is cut such that its base lies on one of the faces of the cube. Which one of the following is correct ? 

(a) VS = VC 

(b) VS = 2VC 

(c) 2VS = 3VC 

(d) 3VS = 4VC

1 Answer

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Best answer

Let the side of the cube be a units. Since sphere is cut off from the cube, radius of the sphere = \(\frac{a}{2}\) units.

∴ Volume of the sphere = \(\frac43\)π \(\big(\frac{a}{2}\big)^3\)

⇒ VS = \(\frac43\) .\(\frac{πa^3}{8}\) =  \(\frac{πa^3}{6}\)cu. units            ...(i) 

Since the cone is cut off from an identical cube, radius of base of cone = \(\frac{a}{2}\) units height of cone = a units.

∴ Volume of cone VC\(\frac13\)π \(\big(\frac{a}{2}\big)^2\). a =  \(\frac13\)π . \(\frac{a^2}{4}\) . a = \(\frac{πa^3}{12}\)     ...(ii) 

From eqn. (i) and (ii), we get VS × \(\frac12\) = VC ⇒ VS = 2 VC.

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