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From a wooden cylindrical block, whose diameter is equal to its height, a sphere of maximum possible volume is carved out. What is the ratio of the utilised wood to that of the wasted wood ? 

(a) 2 : 1 

(b) 1 : 2 

(c) 2 : 3 

(d) 3 : 2

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(a) 2 : 1

Let r be the radius of the cylindrical block, then height of the block = 2r. 

Volume of the block = π(r2) (2r) = 2πr3 

A sphere of maximum possible volume is carved out of the cylindrical block, so the radius of the sphere = r.

∴ Volume of sphere = \(\frac43\)πr3 = Volume of utilised wood 

∴ Volume of wasted wood = 2πr3 – \(\frac43\)πr= \(\frac{2πr^3}{3}\)

∴ Required ratio =  \(\frac43\)πr:  \(\frac23\)πr3 = 2 : 1.

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