Question

If q is the angle (semi-vertical) of a cone of maximum volume and given slant height, then tan q is given by

• A. 2
• B. 1
• C. $\sqrt { 2 }$
• D. ) $\sqrt { 3 }$

Answer 1

Answer: (C)

$\sqrt { 2 }$

Answer 2

Let OB = l, OA = l cos q and AB = l sin q

( 0 £ q £ p/2). Then

V = $\frac { \pi } { 3 }$ (AB)² (OA) = $\frac { \pi } { 3 }$ l³ sin² q cos q

Ž $\frac { \mathrm { dV } } { \mathrm { d } \theta }$ = $\frac { \pi } { 3 }$ l³ sin q (3 cos² q –1)

So from $\frac { \mathrm { dV } } { \mathrm { d } \theta }$ = 0, we get q = 0 or cos q =$\frac { 1 } { \sqrt { 3 } }$. Also V(0) = 0, V (p/2) = 0 and

V $\left( \cos ^ { - 1 } \frac { 1 } { \sqrt { 3 } } \right)$ = $\frac { 2 \pi \ell ^ { 3 } } { 9 \sqrt { 3 } }$

Hence V is maximum when cos q = 1/ $\sqrt { 3 }$ , i.e.,

tan q = $\sqrt { 2 }$