Question

The lateral edge of a regular rectangular pyramid is ' $a$ ' $cm$ long. The lateral edge makes an angle $\alpha$ with the plane of the base. The value of $\alpha$ for which the volume of the pyramid is greatest, is

• A. $\frac{\pi}{4}$
• B. $\sin ^{-1} \sqrt{\frac{2}{3}}$
• C. $\cot ^{-1} \sqrt{2}$
• D. $\frac{\pi}{3}$

Answer 1

Answer: (C)

$h = a \sin \alpha \& x = a \cos \alpha ; x ^2+ h ^2= a ^2 $
$V =\frac{1}{3} y ^2 h =\frac{1}{3} 2 x ^2 h \quad\left(\text { note }: 4 x ^2=2 y ^2 \Rightarrow y ^2=2 x ^2\right) $
$V (\alpha)=\frac{2}{3} a ^2 \cos ^2 \alpha \cdot a \sin \alpha=\frac{2}{3} a ^3 \sin \alpha \cos ^2 \alpha$
$\text { now } \left. V ^{\prime}(\alpha)=0 \Rightarrow \tan \alpha=\frac{1}{\sqrt{2}} ; V _{\max }=\frac{4 \sqrt{3} a ^3}{27}\right]$