This problem involves similar triangles because Sandy's line of sight from her eyes to the top of the flagpole forms one triangle, and the reflection in the mirror forms another triangle.

We know the following:

• $c = 157 \, \text{cm}$ is the height of Sandy's eyes.
• The mirror is placed 2 meters ($a = 200 \, \text{cm}$) away from Sandy.
• The mirror is 6 meters ($b = 600 \, \text{cm}$) away from the flagpole.
• The height of the flagpole is $x \, \text{cm}$.

We can use the properties of similar triangles to set up the ratio:

$\frac{x}{b} = \frac{c}{a}$

Substituting the given values:

$\frac{x}{600} = \frac{157}{200}$

Now, solve for $x$:

$x = \frac{157 \times 600}{200} = \frac{94200}{200} = 471 \, \text{cm}$

Thus, the height of the flagpole is $\boxed{471 \, \text{cm}}$.

Would you like further details on any part of the process?

Here are 5 related questions to consider:

1. How do we know that the triangles are similar?
2. What would happen if the distances to the mirror changed?
3. How could we solve this problem using trigonometry?
4. How would this problem change if Sandy's height were different?
5. What are other real-world applications of similar triangles?

Tip: Always look for opportunities to apply ratios or proportions when dealing with similar triangles in geometry.