31. The city of Sigmagrad has the shape of a square with a side of 5 km (Fig. 5). Streets divide it into blocks, which are squares with a side of 200 m. What is the largest area that can be covered by walking 10 km along the streets of this city and returning to the starting point?

![](https://cdn.mathpix.com/cropped/2024_05_21_7d0959e1f193b6338b84g-12.jpg?height=687&width=678&top_left_y=494&top_left_x=239)

Fig. 5.
31. Suppose we managed to cover the largest area by walking 10 km along the streets of Sigmagrad. Let's prove that we walked in a rectangular path. Let's draw our path on the city map. Let \( AB \) be the "topmost" street we visited, \( DC \) the "bottommost", \( AD \) the "leftmost", and \( BC \) the "rightmost". To go from street \( AB \) to \( DC \) and return, we need to walk along the "vertical" streets for at least \( 2 \cdot AD \), and to go from \( AD \) to \( BC \) and return to the starting point, we need to walk along the "horizontal" streets for at least \( 2 \cdot AB \). Therefore, our path has a length of at least \( 2 \cdot AD + 2 \cdot AB \), which is the perimeter of the rectangle \( ABCD \). If our path were not rectangular, then the enclosing path \( ABCD \) would encompass a larger area with the same or even a smaller perimeter (Fig. 30).

It remains to find the "rectangular" path that encloses the largest area. Let \( a \) be the length of the longer side of the rectangle, and \( b \) the shorter side. Represent \( a \) as: \( a = (2.5 + x) \) km, then \( b = (2.5 - x) \) km, and the area of the rectangle is

\[
S = ab = (6.25 - x^2)
\]

from which it is clear that the area is greater the smaller \( x \) is.

![](https://cdn.mathpix.com/cropped/2024_05_21_7d0959e1f193b6338b84g-45.jpg?height=743&width=465&top_left_y=447&top_left_x=0)

Fig. 30. The smallest possible value of \( x \) in our conditions is 0.1 km, so the largest area that can be covered is \( 6.24 \) km².

|  | \( c \) | 3 | \( \boldsymbol{Ж} \) | \( \kappa \) | \( c \) | \( \sqrt{3} \) |  |  | 10 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
|  |  | 5 | \( \sim \) |  |  | \( \pi \) | \( \kappa \) | \( c \) |  |
|  | \( \mathcal{H} \) | - | \( \underline{c} \) | 3 | ![](https://cdn.mathpix.com/cropped/2024_05_21_7d0959e1f193b6338b84g-45.jpg?height=57&width=50&top_left_y=784&top_left_x=791) | \( \kappa \) | \( c \) | 3 | \( [\bar{m} \) |
|  | \( \kappa \) | \( c \) | 3 | 世 | \( \kappa \) | \( \bar{C} \) | 3 | \( \boldsymbol{\mathcal{K}} \) | \( n \) |
|  | \( = \) | 3 | \( \bar{y} \) | \( \mathbb{K} \) | \( \bar{c} \) | 3 | \( \underset{\xi}{*} \) | \( \mathrm{k} \) | \( C \) |
|  | 3 | \( \overline{\mathcal{H}} \) |  | \( c \) |  | \( \mid \mathcal{W} \) | \( \kappa \) | \( c \) | 3 |
| 3 | ※ | \( \kappa \) | \( C \) | 3 | \( \boldsymbol{\psi} \) | \( \sqrt{\kappa} \) | \( c \) | 3 | \( \vec{H} \) |
| W | \( \kappa \) | \( c \) | 3 | \( \mu \) | \( \kappa \) | \( c \) | 3 | \( \boldsymbol{K} \) | \( \kappa \) |
| 4 | C | 3 | Kl | \( \kappa \) | \( c \) | 3 | \( \boldsymbol{Ж} \) | \( \kappa \) | \( c \) |
|  | 3 | \( \underset{H}{*} \) | \( \kappa \) | \( c \) |  |  |  |  |  |

Fig. 31