5. The edges of the cuboid are in the ratio $4: 3$, the diagonals of the lateral faces are in the ratio $\sqrt{20}: \sqrt{13}$, and the area of the diagonal section is in the ratio 2:1 to the volume of the cuboid. Determine the area and volume of this cuboid.
Solution. Let $a$ and $b$ be the base edges, $d'$ and $d''$ the diagonals of the side faces, $d$ the diagonal of the base, and $h$ the height of the cuboid (see the diagram on the right). From the given dimensions, we get the proportions $a: b=4: 3, d': d''=\sqrt{20}: \sqrt{13}, d h: a b h=2: 1$. From this, $d'^2: d''^2=20: 13$, that is,

![](https://cdn.mathpix.com/cropped/2024_06_05_e0d95f5510204c7c0f0fg-5.jpg?height=363&width=407&top_left_y=761&top_left_x=1075)

$$
\left(a^{2}+h^{2}\right):\left(b^{2}+h^{2}\right)=20: 13 \text { and } d: a b=2: 1
$$

From $d: a b=2: 1$ it follows that $\sqrt{a^{2}+b^{2}}: a b=2: 1$, i.e., $\left(a^{2}+b^{2}\right): a^{2} b^{2}=4: 1$. Now, from

$$
a: b=4: 3 \text { and }\left(a^{2}+b^{2}\right): a^{2} b^{2}=4: 1
$$

we have $4 b=3 a$ and $a^{2}+b^{2}=4 a^{2} b^{2}$, so by substituting $b=\frac{3}{4} a$, after simplification we get $a^{2}=\frac{25}{36}$, i.e., $a=\frac{5}{6}$. Further, $b=\frac{3}{4} a=\frac{5}{8}$. Now, substituting into $\left(a^{2}+h^{2}\right):\left(b^{2}+h^{2}\right)=20: 13$ we get

$$
\left(\frac{25}{36}+h^{2}\right):\left(\frac{25}{64}+h^{2}\right)=20: 13
$$

from which $h^{2}=\frac{25}{144}$, i.e., $h=\frac{5}{12}$.

Finally, $V=a b h=\frac{125}{576}$.