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In this problem, Nancy is trying to observe a man behind her using a small plane mirror. The question asks for the maximum distance Nancy can place the mirror away from her eyes in order to see the full image of the man behind her. Here's how we approach it:

### Given dataIn this problem, Nancy is trying to observe a man behind her using a small plane mirror. The question asks for the maximum distance Nancy can place the mirror away from her eyes in order to see the full image of the man behind her. Here's how we approach it:

### Given 
- The man is 2.4 m behind Nancy.
- The man’s height is 1.8 m.
- The mirror is 10 cm long.
- Nancy’s eyes are 1.5 m above the ground.
- Nancy holds the mirror vertically.

### Key Concepts:
To see the full image of the man, the height of the mirror must be at least half of the height of the man. This is based on the law of reflection and the geometry of plane mirrors. In this case, the minimum height of the mirror \( h_m \) required to view the full image is:

\[
h_m = \frac{1}{2} \times \text{man's height} = \frac{1}{2} \times 1.8 = 0.9 \, \text{m}
\]

The mirror available is only 10 cm (0.1 m) tall, which is much smaller than the required 0.9 m. Therefore, Nancy needs to place the mirror closer to her eyes to make sure that the 10 cm mirror can show the full image of the man. This can be determined by using similar triangles formed by Nancy’s line of sight.

The situation is described using similar triangles because the angles of reflection must be the same for both the man and the image in the mirror. 

#### Formula to use:
\[
\frac{H_m}{h_m} = \frac{d_m}{d}
\]
Where:
- \( H_m = 1.8 \, \text{m} \) (height of the man)
- \( h_m = 0.1 \, \text{m} \) (height of the mirror)
- \( d_m = 2.4 \, \text{m} \) (distance of the man from Nancy)
- \( d \) is the distance from Nancy’s eyes to the mirror, which we want to find.

By solving this, we can find the maximum distance Nancy can place the mirror to see the full image of the man.

Let me calculate that for you.The maximum distance between Nancy and the mirror for her to see the whole image of the man is approximately **13.3 cm**.

This corresponds to option **A** in the given choices.

Would you like further details or clarifications?

Here are 5 related questions to think about:
1. How does the size of the mirror impact what Nancy sees?
2. Why does Nancy need only half the mirror height to see the full image of the man?
3. What if the mirror were curved—how would the image change?
4. Can Nancy see the full image if the man is closer or farther away?
5. How would Nancy's position affect the clarity of the image?

### Tip:
Always remember that in a plane mirror, the size of the image is the same as the object, and the distance of the object from the mirror is equal to the distance of the image behind the mirror.

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. Assume Nancy holds the mirror vertically. What is the maximum distance between Nancy and the mirror for her to see the whole image of the man?

Discover how to calculate the maximum distance a plane mirror should be from the viewer to see a full reflection of an object behind. This problem involves geometry, proportions, and optics, suitable for high school students.

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