The square-shaped square is being reconstructed with a side length of 20 meters. If the entire square were to be paved with cheaper light-colored paving stones, the material costs would be 100,000 Kč. If the entire square were to be covered with more expensive dark paving stones, the material costs would be 300,000 Kč. However, the architect has proposed a light-colored square section in the central part of the square, which will be bordered by a 2-meter-wide strip of dark paving stones, and the outer part of the square will be the same light-colored paving stones as in the center (see image). According to this proposal, the material costs for the dark parts will be the same as for the total light parts.

![](https://cdn.mathpix.com/cropped/2024_04_17_7af2ed77a4c56d0b6466g-2.jpg?height=580&width=556&top_left_y=658&top_left_x=750)

## Determine:

a) how much the material for paving the square according to this project will cost,

b) what the side length of the light square in the central part of the square will be.

(L. Šimůnek)
a) Light paving is three times cheaper than dark paving. The costs for the light and dark parts of the square are the same, so the light area is three times larger than the dark area. The dark area thus forms one quarter of the square, and the cost for it is 300000 : $4=75000$ (Kč). The costs for the light part are the same. Therefore, the paving of the entire square according to the given project costs

$$
2 \cdot 75000=150000 \text { (Kč). }
$$

b) The entire square has an area of $20 \cdot 20=400\left(\mathrm{~m}^{2}\right)$. From the previous part, we know that the darker area forms one quarter of the square, i.e., $400: 4=100\left(\mathrm{~m}^{2}\right)$. This area can be expressed as the sum of four identical squares and four identical rectangles, see the image.

![](https://cdn.mathpix.com/cropped/2024_04_17_7af2ed77a4c56d0b6466g-2.jpg?height=537&width=554&top_left_y=2090&top_left_x=751)

Each of the mentioned squares has an area of $2 \cdot 2=4\left(\mathrm{~m}^{2}\right)$, so the remaining part of the dark area has an area of $100-4 \cdot 4=84\left(\mathrm{~m}^{2}\right)$. The area of each of the mentioned rectangles is therefore $84: 4=21\left(\mathrm{~m}^{2}\right)$. One side of the rectangle measures $2 \mathrm{~m}$, so the length of the other side is

$$
21: 2=10.5(\mathrm{~m})
$$

and this is also the length of the side of the light square in the central part of the square.

Grading proposal. 2 points for task a), of which 1 point for the explanation; 4 points for task b), of which 3 points for the explanation (e.g., 1 point for the area of the gray area, 1 point for the idea to separate the squares in the corners, 1 point for the area of the auxiliary rectangle).