\section*{Problem 2 - 151222}

Given is a pyramid whose base is a rectangle with area \(Q\). Two of the side faces are perpendicular to the base; the other two form angles of size \(\alpha\) and \(\beta\) with the base.

Determine the volume of the pyramid in terms of \(Q, \alpha\), and \(\beta\).
}

Let \(A B C D S\) be a pyramid whose base \(A B C D\) is a rectangle with area \(Q\) and whose apex is \(S\).

The side faces perpendicular to the base share a common edge of the pyramid, as otherwise they would belong to parallel planes, which is not possible in a pyramid. This edge of the pyramid perpendicular to the base is \(S D\).

Therefore, for the side faces \(B C S\) and \(A B S\) that are not perpendicular to the base, the inclination angles are equal to the angles \(\angle S A D\) and \(\angle S C D\). Additionally, the length of \(S D\) is equal to the height \(h\) of the pyramid.

Without loss of generality, let \(\angle S A D = \alpha\) and \(\angle S C D = \beta\). Furthermore, we have

\[
\begin{aligned}
h = S D & = A D \cdot \tan \alpha \\
& = C D \cdot \tan \beta \\
Q & = A D \cdot C D
\end{aligned}
\]

From (1) and (2), by multiplying and then substituting (3) and taking the square root, we get:

\[
h^{2} = A D \cdot C D \cdot \tan \alpha \cdot \tan \beta \quad ; \quad h = \sqrt{Q \cdot \tan \alpha \cdot \tan \beta}
\]

Thus, the volume \(V\) of the pyramid is:

\[
V = \frac{1}{3} Q h = \frac{1}{3} Q \sqrt{Q \cdot \tan \alpha \cdot \tan \beta}
\]

\section*{Adapted from \([5]\)}