## 26. Math Puzzle $7 / 67$

On the sports field, a rectangle is marked out. If one of the rectangle's sides is extended by 2 meters and the other is shortened by 3 meters, a square is formed whose area is $1 \mathrm{~m}^{2}$ smaller than that of the original rectangle.

How large is the area of the rectangle?

From the problem statement, the following equations can be derived: $(x+2)(y-3)=x y-1$ and $x+2=y-3$. Therefore, if we set $x=y-5$: $x=5, y=10$ and an area of $50 \mathrm{~m}^{2}$.

![](https://cdn.mathpix.com/cropped/2024_06_06_69be9c4355ed0dc2557cg-12.jpg?height=300&width=449&top_left_y=747&top_left_x=815)

The new television tower on Berlin's Alexanderplatz will reach a height of $353 \mathrm{~m}$ upon completion. Its total mass will then be approximately $19700 \mathrm{t}$.

How large would a scale and detail-accurate model of the television tower be if it were made from the same materials and had a mass of $1 \mathrm{~kg}$?
Since in a scale reduction of the height of the TV tower, the other two dimensions also shorten in the same scale, it follows that:

$$
\frac{x}{353}=\sqrt[3]{\frac{1}{19700000}} \rightarrow x=\frac{353}{270} \approx 1.307 \mathrm{~m}
$$