Question

The diameter of a solid mettalic right circular cylinder is equal to its height. After culting out the largest possible solid sphere S from this cylinder, the remaining material is recast to form a solid sphere S₁. What is the ratio of the radius of the sphere S to that of the sphere S₁. 

(a) \(1:2^{\frac13}\)

(b) \(2^{\frac13}:1\)

(c) \(2^{\frac13}:3^{\frac13}\)

(d) \(3^{\frac12}:2^{\frac12}\)

Answer 1

(b) \(2^{\frac13}:1\)

Let the height of the cylinder = h cm. 

Then, radius of cylinder = radius of sphere = \(\frac{h}{2}\) cm.

∴ Volume of remaining material 

= Volume of cylinder – Volume of sphere 

= \(π\big(\frac{h}{2}\big)^2h-\frac43π\big(\frac{h}{2}\big)^3=\frac{πh^3}{4}-\frac{πh^3}{6}=\frac{πh^3}{12}\)

Let the recasted solid sphere have radius R cm. Then, volume of recasted sphere = Volume of remaining material.

⇒ \(\frac43πR^3 = \frac{πh^3}{12}\) ⇒ R³ = \(\frac{h^3}{16}\) ⇒ R = \(\frac{h}{2.(2^{\frac13})}\)

∴ Required ratio = r : R = \(\frac{h}{2}\) : \(\frac{h}{2.(2^{\frac13})}\) = \(2^{\frac13}:1\).