Volume of a sphere varies directly with cube of its radius. Three spheres with radii in the ratio 3 : 4 : 5 are melted to form a large sphere. What is the ratio of radius of smallest sphere to the new sphere formed.
Explanation:
Given, volume is directly proportional to cube of radius.
⇒ V = kr3
Let the radii of the original cubes be 3r, 4r and 5r and that of the larger cube be R.
Since the three original cubes are melted to form the larger cube, total volume of original three cubes will be same as volume of the larger cube.
∴ k(3r)3 + k(4r)3 + k(5r)3 = k(R)3
⇒ 27r3 + 64r3 + 125r3 = R3
⇒ 216r3 = R3
⇒ R = 6r
∴ Radius of smallest sphere : Radius of the new sphere = 3r : 6r = 1 : 2
Hence, option (a).
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