Question

The lateral edge of a regular hexagonal pyramid is 1 cm. If the volume is maximum, then its height is

• A. $\frac{1}{\sqrt{3}}$
• B. $\frac{6}{\sqrt{3}}$
• C. $\frac{1}{6}$
• D. $\frac{1}{3}$

Answer 1

The correct option is A $\frac{1}{\sqrt{3}}$

$△AOP$ is right angled
$\Rightarrow$ $x^2+h^2=1;$
Volume, $V=\frac{\sqrt{3}}{2}{x}^{2}h=\frac{\sqrt{3}}{2}h(1-h^2)$
$V^{\prime }\left(h\right)=\frac{\sqrt{3}}{2}(1-3h^2)$
For $V^{\prime }\left(h\right)=0$
$\Rightarrow h=\frac{1}{\sqrt{3}}$
$\{\because h\text{cannot be negative}\}$
$V^{\prime \prime }\left(\frac{1}{\sqrt{3}}\right)=-3<0$
$\Rightarrow$ at $h=\frac{1}{\sqrt{3}}$ Volume is maximum.