What is the value of $\alpha$ for which the volume of the regular rectangular pyramid is greatest?
Solution 1
You have not only $h=a\sin\alpha$, but also $b=\sqrt2 a\cos\alpha$, where $b$ is the side of the base (a regular rectangular pyramid has a square base). Plugging these into the volume formula, we get: $$ V={2\over3}a^3\sin\alpha\cos^2\alpha. $$ To find the maximum $V$ we just need to find the solutions of equation $dV/d\alpha=0$, which boils down to: $$ \cos^2\alpha=2\sin^2\alpha, \quad\hbox{that is:}\quad \tan\alpha={1\over\sqrt2}, \quad \alpha\approx35.26°. $$
Solution 2
The main reason this question looks confusing to me is that choosing $\alpha$ alone does not determine the volume, you also need to optimize $l$ and $b$. As half the diagonal of the base rectangle sits in a rightangeled triangle with $a$ and $h$, the constraint there is $$\frac{1}{4}l^2 + \frac{1}{4}b^2 + h^2 = a^2$$ or in other words $$l^2 + b^2 = 4(1\sin^2(\alpha))\cdot a^2 = 4\cos^2(\alpha)\cdot a^2.$$ Now you can actually break the optimization problem up into two parts: For fixed $\alpha$, you can choose $l,b$ such that their product $P = P(\alpha)$ is maximal, then you optimize $V =\frac{1}{3} a \cdot P(\alpha)\cdot \sin(\alpha)$.
For the first part, note that $l\cdot b \leq \frac{1}{2} (l^2 + b^2)$ as $l^2 +b^2  2l\cdot b = (lb)^2 \geq 0$; so $l\cdot b$ is maximal if $l = b$, and in this case we have $$l \cdot b = \frac{1}{2} (l^2 + b^2) = 2\cos^2(\alpha)\cdot a =: P(\alpha)$$
So your pyramid has quadratic base and Volume $$V = \frac{2}{3}a^2 \cdot(\cos^2(\alpha)\sin(\alpha)).$$
So you're left with optimizing $$\cos^2(\alpha)\sin(\alpha) = \sin(\alpha)  \sin^3(\alpha)$$ which you can do by taking the derivative w.r.t $\alpha$ and search for zeros.
user1442
Updated on August 01, 2022Comments

user1442 over 1 year
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.What is the value of $\alpha$ for which the volume of the regular rectangular pyramid is greatest?
Since the volume of the rectangular pyramid is $V=\frac{lbh}{3}$,where $l$ is
the length of the base of the pyramid,$b=$width of the base of the pyramid,
$h=$ height of the pyramid.and $\sin \alpha=\frac{h}{a}$,where $a$ is the lateral edge(as given in the question.)$\Rightarrow h=a\sin\alpha$
So $V=\frac{l\times b \times a\times\sin\alpha}{3}$
What should i do to maximize the volume? 
user1442 about 8 yearsWhy is it so that in a constrained problem,a product of three numbers is maximum when they are all equal.Can you give me a link where i can read the proof of this concept.And how is present case is a case of constrained case.@Narasimham

Han de Bruijn over 6 yearsIf you replace your last three lines by the last two lines of Aretino's answer (or the other way around) then both answers together are perfect.

Han de Bruijn over 6 yearsIf you augment your answer with the first 13 lines of Johann's answer (or the other way around) then both answers together are perfect.

Intelligenti pauca over 6 years@HandeBruijn To me, REGULAR rectangular pyramid means it has a SQUARE base.

Intelligenti pauca over 6 yearsJust edited my answer to make that clear.