Question

A lateral edge of a regular rectangular pyramid is a long. The lateral edge makes an angle $\alpha$ with the plane of the base. For what $\alpha$ is the volume of the pyramid the greatest?

Answer 1

Volume of rectangular pyramid, $V=\frac{lbh}{3}$

$\sin \alpha =\frac{h}{a}\Rightarrow h=a\sin \alpha$

& $l=b=2a$

Then, $V=\frac{4a^3\sin \alpha }{3}\Rightarrow \frac{dV}{d\alpha }=\frac{4a^3\cos \alpha }{3}$

For maxima, $\frac{d^{2}V}{d{\alpha }^2}=-\frac{4a^3\sin \alpha }{3}<0$

$\therefore \frac{dV}{d\alpha }=0\Rightarrow \frac{4a^3\cos \alpha }{3}=0\Rightarrow 4a^3\cos \alpha =0$

$4a^3\ne 0$ then, $\cos \alpha =0\therefore \alpha =\frac{\pi }{2}$ or$\frac{3\pi }{4}$