https://r2.erweima.ai/mathbot/image/1724113648948kbjs40kl.jpg

Let's analyze the problem presented in the image and solve it step by step.

### Given Information:
1. The task involves finding the maximum speed \((v_{\text{max}})\) a train can have while maintaining a safe distance between two stops, using the given speed formula related to distances \(S_1\) and \(S_2\).

2. The equation provided is:
   \[
   \sqrt{2} \cdot 20 \geq \sqrt{v^2 + 121}
   \]
   We need to solve for \(v\) to determine the maximum speed.

### Steps to solve:

1. **Simplify the Inequality:**
   The inequality is given as:
   \[
   \sqrt{2} \times 20 \geq \sqrt{v^2 + 121}
   \]
   First, simplify the left side:
   \[
   \sqrt{2} \times 20 = 20\sqrt{2}
   \]
   Now, we have:
   \[
   20\sqrt{2} \geq \sqrt{v^2 + 121}
   \]

2. **Square Both Sides:**
   To eliminate the square root, square both sides of the inequality:
   \[
   (20\sqrt{2})^2 \geq v^2 + 121
   \]
   Simplify further:
   \[
   400 \times 2 \geq v^2 + 121
   \]
   \[
   800 \geq v^2 + 121
   \]

3. **Isolate \(v^2\):**
   Subtract 121 from both sides:
   \[
   800 - 121 \geq v^2
   \]
   \[
   679 \geq v^2
   \]
   \[
   v^2 \leq 679
   \]

4. **Solve for \(v\):**
   Take the square root of both sides:
   \[
   v \leq \sqrt{679}
   \]
   Approximating the square root:
   \[
   v \leq \sqrt{679} \approx 26.05 \, \text{m/s}
   \]

### Filling in the blanks:
1. The solution to the inequality is \( v \leq \sqrt{679} \).
2. The maximum speed that satisfies the condition is approximately \( 26.05 \, \text{m/s} \).

Would you like more details or have any other questions? Here are some related questions you might find helpful:

1. How do you solve inequalities that involve square roots?
2. What are some common applications of quadratic inequalities in physics?
3. How does squaring both sides of an equation affect the solution?
4. What methods can be used to approximate square roots?
5. What role does the safety distance play in calculating the maximum speed?

**Tip:** Always check the results of inequalities, especially when dealing with squared terms, to ensure the solutions are valid within the given context.

Learn how to solve an inequality involving square roots step-by-step. This problem covers key mathematical concepts such as inequalities and square roots, providing a detailed solution and approximation for the maximum speed in a given context.

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