Explanation: To solve this problem, we need to understand the relationship between the time taken by two individuals to complete their journeys after crossing each other and the ratio of their speeds. When two individuals, A and B, start walking towards each other from two different points and cross each other, the time taken by each to complete their journey after crossing is inversely proportional to the square of their speeds.
Let's denote the speed of A as \( v_A \) and the speed of B as \( v_B \). The time taken by A to complete the journey after crossing B is 361 hours, and the time taken by B to complete the journey after crossing A is 289 hours. According to the principle mentioned, the ratio of the speeds of A and B is given by the inverse square root of the times taken after crossing each other.
Mathematically, this can be expressed as:
\[
\frac{v_A}{v_B} = \sqrt{\frac{t_B}{t_A}}
\]
where \( t_A \) is the time taken by A after crossing B, and \( t_B \) is the time taken by B after crossing A.
Substituting the given values:
\[
\frac{v_A}{v_B} = \sqrt{\frac{289}{361}}
\]
We know that:
\[
\sqrt{289} = 17 \quad \text{and} \quad \sqrt{361} = 19
\]
Thus:
\[
\frac{v_A}{v_B} = \frac{17}{19}
\]
Therefore, the ratio of the speed of A to the speed of B is 17:19.
This problem highlights the importance of understanding the relationship between time and speed in relative motion problems. The key is to recognize that the time taken after crossing each other is inversely proportional to the square of the speeds, which allows us to find the ratio of the speeds directly from the given times.