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)

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