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In this problem, we need to determine the maximum distance between Nancy and the small mirror she uses to see the full image of the man behind her. Here's a breakdown of the information provided:

- The man is **2.4 m** behind Nancy.
- The man is **1.8 m** tall.
- Nancy holds a small mirror of length **10 cm** vertically.
- Nancy's eyes are **1.5 m** above the ground.
- We assume the mirror is aligned vertically.

To understand the optical principle at work, we recall that in order to view the full image of the man in a plane mirror, the height of the mirror required is only half the height of the man due to symmetry. This means the height of the image seen in the mirror is directly proportional to the height of the man in reality.

### Key Concept:
For Nancy to see the full image of the man, the vertical length of the mirror needs to allow the entire image of the man to be visible, both above and below the level of her eyes. This involves ray tracing and understanding the reflection angles.

To find the maximum distance Nancy can be from the mirror while still seeing the full image of the man, we apply geometrical optics principles, which include reflection laws and the relationship between object distance, image size, and mirror size.

Let me calculate this step-by-step and provide a precise solution.The maximum distance Nancy can be from the mirror to see the entire image of the man is **12 cm**. This value doesn't match the given options in the problem. However, based on typical mirror viewing geometry, other factors such as ray angles might be involved that suggest the actual answer could be slightly more than this, leaning towards one of the options.

Would you like further clarification, or do you have any other questions?

Here are 5 related questions to expand your understanding:

1. How does the distance between Nancy and the mirror affect the size of the reflected image?
2. Why does Nancy only need a mirror half the size of the man to see his full image?
3. How would the answer change if Nancy's eye level was different?
4. What optical principles are involved in determining image size in a plane mirror?
5. How does the reflection angle in a plane mirror impact the clarity of the image?

**Tip**: When dealing with plane mirrors, remember that the image formed is always the same size as the object, but you only need half the mirror height to see the full image!

Nancy tries to use a small plane mirror to observe a man 2.4 m behind her. The man is 1.8 m tall. The mirror is 10 cm long. Nancy's eyes are 1.5 m above the ground. What is the maximum distance between Nancy and the mirror for her to see the whole image of the man?

Learn how to calculate the maximum distance between Nancy and a mirror for her to observe the full image of a person 2.4 meters behind her. This problem applies geometrical optics and reflection principles, including the symmetry of images in plane mirrors. Suitable for high school students.

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