5. Find the height of the pyramid if, by cutting it only along the lateral edges and unfolding the lateral faces onto the plane of the base, outside of it, you get a square with a side of 18. If it is impossible to get such a square, explain why.

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# Solution:

![](https://cdn.mathpix.com/cropped/2024_05_06_e4d6b0c7cc0369892c18g-5.jpg?height=674&width=1180&top_left_y=98&top_left_x=128)

The figure shows such a net. If we flip the pyramid and place it on the face $A B C$, it is easy to calculate the volume of the pyramid $V=\frac{1}{3} \cdot \frac{1}{2} \cdot 9 \cdot 9 \cdot 18=3^{5}$. We determine the area of the base $B D C$ as the difference between the area of the square and the areas of three triangles. $18^{2}-\frac{1}{2}(9 \cdot 9+9 \cdot 18+9 \cdot 18)=\frac{3^{5}}{2}$

We determine the volume of the pyramid in two ways. $V=\frac{1}{3} H \cdot \frac{3^{5}}{2}=3^{5}$, where H is the height of the pyramid we are looking for. From the equality, we find that $H=6$.

## Answer: 6

## Recommendations for checking.

The net of the pyramid is correctly drawn - 4 points.