Problem 1419

1. Calculate the area of the large circle:
The radius of the large circle is given as $2$ meters. The area $A$ of a circle is given by the formula:
$$A = \pi r^2$$
Substituting the radius:
$$A_{\text{large}} = \pi (2)^2 = 4\pi \text{ square meters}$$

2. Calculate the area of one small circle:
The radius of each small circle is $1$ meter. Using the same formula for the area of a circle:
$$A_{\text{small}} = \pi (1)^2 = \pi \text{ square meters}$$
Since there are four small circles, the total area of the small circles is:
$$4A_{\text{small}} = 4\pi \text{ square meters}$$

3. Calculate the area of the dark grey region:
The dark grey region is the area of the large circle minus the area of the four small circles plus the area of the light grey region (since the small circles overlap and we subtract too much).
$$A_{\text{dark}} = A_{\text{large}} - 4A_{\text{small}} + A_{\text{light}}$$
Substituting the known areas:
$$A_{\text{dark}} = 4\pi - 4\pi + A_{\text{light}}$$
Simplifying, we find:
$$A_{\text{dark}} = A_{\text{light}}$$

4. Calculate the area of the light grey region:
To find the area of the light grey region, consider the overlapping area of two small circles. The overlapping region can be found by calculating the area of the sector minus the area of the triangle formed by the radii and the chord.
$$\text{Area of sector} = \frac{90^\circ}{360^\circ} \pi (1)^2 = \frac{\pi}{4}$$
$$\text{Area of triangle} = \frac{1 \times 1}{2} = \frac{1}{2}$$
Therefore, the area of one overlapping region is:
$$\text{Area of one overlapping region} = \frac{\pi}{4} - \frac{1}{2}$$
Since there are four such overlapping regions:
$$A_{\text{light}} = 4 \left( \frac{\pi}{4} - \frac{1}{2} \right) = \pi - 2$$

5. Calculate the area of the medium grey region:
The medium grey region is the area of the four small circles minus the area of the light grey region:
$$A_{\text{medium}} = 4A_{\text{small}} - A_{\text{light}}$$
Substituting the known areas:
$$A_{\text{medium}} = 4\pi - (\pi - 2) = 4\pi - \pi + 2 = 3\pi + 2$$

6. Calculate the cost of painting the figure:
The cost of painting each region is given by:
- Dark grey region: $30 \text{ kr. per square meter}$
- Medium grey region: $20 \text{ kr. per square meter}$
- Light grey region: $10 \text{ kr. per square meter}$

Therefore, the total cost is:
$$\text{Total cost} = 30A_{\text{dark}} + 20A_{\text{medium}} + 10A_{\text{light}}$$
Substituting the areas:
$$\text{Total cost} = 30(\pi - 2) + 20(3\pi + 2) + 10(\pi - 2)$$
Simplifying:
$$\text{Total cost} = 30\pi - 60 + 60\pi + 40 + 10\pi - 20$$
$$\text{Total cost} = 100\pi - 40$$

The final answer is $\boxed{100\pi - 40 \text{ kr.}}$