1. The base of a right quadrilateral prism is a rhombus with an area of $\frac{2}{3} k^{2}$. The smaller diagonal section of the prism is a square with an area of $k^{2}$. a) Calculate the surface area and volume of the prism expressed in terms of $k$.

b) Determine $k$ if the numerical values of the surface area and volume of the prism are equal.
Solution. From the area of the smaller diagonal section, we conclude that the smaller diagonal of the rhombus and the height have a length of $k$ (see the diagram on the right). From the area of the rhombus, we obtain the unknown diagonal:

$$
\frac{d k}{2}=\frac{2}{3} k^{2}, \text { i.e., } d=\frac{4}{3} k
$$

The side of the rhombus is calculated from the right-angled triangle in the diagram on the left. We have

![](https://cdn.mathpix.com/cropped/2024_06_05_e2b4ba6b48251e03981fg-3.jpg?height=346&width=310&top_left_y=726&top_left_x=1152)
shadowed

![](https://cdn.mathpix.com/cropped/2024_06_05_e2b4ba6b48251e03981fg-3.jpg?height=209&width=332&top_left_y=1150&top_left_x=324)

$a$

$$
a^{2}=\left(\frac{k}{2}\right)^{2}+\left(\frac{2 k}{3}\right)^{2}, \text { i.e., } a=\frac{5}{6} k
$$

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

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

b) We have $V=P$, i.e., $\frac{2}{3} k^{3}=\frac{14}{3} k^{2}$, from which we obtain $k=7$.