3. The base of a right quadrilateral prism is a rhombus with an area of $\frac{2}{3} k^{2}$, $k>0$. The smaller diagonal section of the prism is a square with an area of $k^{2}$.

a) Express the surface area and volume of the prism in terms of $k$.

b) For what value of $k$ are the numerical values of the surface area and volume of the prism equal to each other?
Solution. From the fact that the smaller diagonal cross-section of the prism is a square with area $k^{2}$, it follows that the height of the prism $H=k$ and the smaller diagonal of its base $d_{1}=k$. The base of the prism is a rhombus with area $B=\frac{2}{3} k^{2}$, from which we get $\frac{2}{3} k^{2}=\frac{d_{1} \cdot d_{2}}{2}$, or $\frac{2}{3} k^{2}=\frac{k \cdot d_{2}}{2}$. From

![](https://cdn.mathpix.com/cropped/2024_06_05_3cde5d119ebb42d699c4g-7.jpg?height=447&width=469&top_left_y=959&top_left_x=972)
the last equation, we get the larger diagonal of the rhombus $d_{2}=\frac{4}{3} k$. From the right triangle $\triangle A B O$ we get

$$
a^{2}=\left(\frac{d_{1}}{2}\right)^{2}+\left(\frac{d_{2}}{2}\right)^{2}, \text { i.e. } a^{2}=\left(\frac{k}{2}\right)^{2}+\left(\frac{4 k}{6}\right)^{2}
$$

from which we get $a=\frac{5}{6} k$.

a) For the volume and surface area of the prism, we get

$$
V=B \cdot H=\frac{2}{3} k^{2} \cdot k=\frac{2}{3} k^{3} \text { and } P=2 B+4 a k=2 \cdot \frac{2}{3} k^{2}+4 \cdot \frac{5}{6} k^{2}=\frac{14}{3} k^{2}
$$

b) From the condition that the numerical values of the area and volume of the prism are equal, we get $\frac{2}{3} k^{3}=\frac{14}{3} k^{2}$, from which $k=7$.