This problem involves related rates and can be analyzed geometrically. The key is to determine the rate at which the length of the kite string ($s$) is increasing, given the height of the kite and the combined horizontal motion.

Step-by-step Solution:

Step 1: Define the problem geometry

1. The height of the kite is constant at $h = 30 \, \text{m}$.
2. Budek is running in one direction at $v_{\text{B}} = 200 \, \text{m/min}$.
3. The wind carries the kite horizontally in the opposite direction at $v_{\text{K}} = 40 \, \text{m/min}$.
4. The total horizontal distance $x$ between Budek and the kite increases at a rate of: $\frac{dx}{dt} = v_{\text{B}} + v_{\text{K}} = 200 + 40 = 240 \, \text{m/min}.$

Step 2: Relate the length of the string ($s$) to the geometry

The string forms the hypotenuse of a right triangle, where: $s^2 = x^2 + h^2.$

Differentiating both sides with respect to time $t$: $2s \frac{ds}{dt} = 2x \frac{dx}{dt}.$

Simplify: $s \frac{ds}{dt} = x \frac{dx}{dt}.$

Solve for $\frac{ds}{dt}$, the rate at which the string is paying out: $\frac{ds}{dt} = \frac{x}{s} \frac{dx}{dt}.$

Step 3: Calculate $x$, $s$, and rates at $t = 10 \, \text{s}$

1. Convert $\frac{dx}{dt}$ into meters per second: $\frac{dx}{dt} = 240 \, \text{m/min} = 4 \, \text{m/s}.$

2. The horizontal distance $x$ after 10 seconds: $x = \frac{dx}{dt} \times t = 4 \, \text{m/s} \times 10 \, \text{s} = 40 \, \text{m}.$

3. The length of the string $s$ is: $s = \sqrt{x^2 + h^2} = \sqrt{40^2 + 30^2} = \sqrt{1600 + 900} = \sqrt{2500} = 50 \, \text{m}.$

Step 4: Calculate $\frac{ds}{dt}$

Using the formula: $\frac{ds}{dt} = \frac{x}{s} \frac{dx}{dt},$ substitute $x = 40 \, \text{m}$, $s = 50 \, \text{m}$, and $\frac{dx}{dt} = 4 \, \text{m/s}$: $\frac{ds}{dt} = \frac{40}{50} \cdot 4 = 0.8 \cdot 4 = 3.2 \, \text{m/s}.$

Final Answer:

The rate at which the string is paying out after 10 seconds is: $\boxed{3.2 \, \text{m/s}}$

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Would you like further details or clarification? Here are related questions to explore:

1. What happens if Budek stops running after 10 seconds?
2. How does the string length change if the kite's height increases over time?
3. How would the result change if Budek and the kite moved in the same direction?
4. What is the string's rate of change if Budek changes direction halfway?
5. Can this problem be solved using parametric equations instead?

Tip: Always ensure the units are consistent when solving related rates problems.