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Question

The maximum volume (in cu.m) of the right circular cone having slant height 3 m is:

A
6π
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B
33π
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C
23π
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D
43π
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Solution

The correct option is C 23π
Given :
Slant height of the cone, L=3 m.
Let the radius be 'r' and height be 'h'.

Volume of a right circular cone,
V=13πr2h
V=13π(L2h2)h [L2=r2+h2]V=13π(9h2) h

Differentiating V w.r.t. h and equating to 0, we get
dVdh=13π(93h2)=0h=±3
As height cannot be negative so, h=3 m
Again differentiating V w.r.t. h,
d2Vdh2=2πh<0 at h=3 m
Maximum volume occurs at h=3.

Hence, the maximum volume of the cone,
V=13π(9h2) h =13π(9(3)2) 3=23π m3

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