5. The area of a regular quadrilateral pyramid is $5 a^{2}$, where $a$ is the length of the side of the base of the pyramid.

a) Express the volume of this pyramid as a function of $a$.

b) Calculate the volume of the pyramid if $a=6 dm$.
Solution. a) For the area of the pyramid, we have

$$
P=a^{2}+4 \cdot \frac{a h}{2}=a^{2}+2 a h
$$

so $a^{2}+2 a h=5 a^{2}$, from which we get $h=2 a$. Furthermore, from the Pythagorean theorem, we get

$$
H^{2}=h^{2}-\left(\frac{a}{2}\right)^{2}=\frac{15 a^{2}}{4}
$$

so $H=\frac{a}{2} \sqrt{15}$.

![](https://cdn.mathpix.com/cropped/2024_06_05_6a3de2303cb9ec4e16a2g-3.jpg?height=512&width=659&top_left_y=1316&top_left_x=805)

According to this,

$$
V=\frac{B H}{3}=\frac{1}{3} a^{2} \cdot \frac{a}{2} \sqrt{15}=\frac{\sqrt{15}}{6} a^{3}
$$

b) For $a=6 \, \text{dm}$, we have:

$$
V=\frac{\sqrt{15}}{6} \cdot 6^{3}=36 \sqrt{15} \, \text{dm}^{3}
$$