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

To solve this problem, we need to determine the maximum distance between Nancy and the mirror such that she can see the entire reflection of the man behind her. Here are the key points and steps we'll follow to find the solution:

1. **Understand the Situation**:
   - Nancy wants to observe a man who is 2.4 meters behind her.
   - The man is 1.8 meters tall.
   - Nancy's eyes are 1.5 meters above the ground.
   - The mirror she is using is 10 cm (0.1 meters) long.
   - We need to find the maximum distance \( d \) between Nancy and the mirror so she can see the entire image of the man in the mirror.

2. **Geometry of Reflection**:
   - For Nancy to see the entire image of the man, the mirror must reflect the light from the top of the man's head and the bottom of his feet to Nancy's eyes.
   - We will use the principles of similar triangles and the law of reflection to establish the relationship between the heights, distances, and the mirror's size.

3. **Calculation of Minimum Mirror Size**:
   - The height of the mirror required to see the entire man is based on the principle that the angle of incidence equals the angle of reflection. The required size of the mirror to see the entire height of the man can be found by the concept that the mirror's size needs to be half the height of the man as seen by Nancy's eyes in reflection.
   - Since the man is 1.8 m tall and Nancy's eyes are 1.5 m from the ground, we need to compute the image distance from Nancy's eyes.

4. **Setting Up the Geometry**:
   - The distance from Nancy to the mirror is \( d \), and the total distance from the man to the mirror is \( 2.4 + d \).
   - For Nancy to see the top and bottom of the man in the mirror, the height of the mirror necessary \( h_m \) can be given by similar triangles:

   \[
   \frac{h_m}{d} = \frac{1.8 \, \text{m}}{2.4 + d}
   \]

   We know that the mirror height \( h_m \) is 0.1 meters.

5. **Calculating \( d \) (Distance)**:
   - Plugging the known values into the equation:

   \[
   \frac{0.1}{d} = \frac{1.8}{2.4 + d}
   \]

   - Cross-multiply and solve for \( d \):

   \[
   0.1 \times (2.4 + d) = 1.8d
   \]

   \[
   0.24 + 0.1d = 1.8d
   \]

   \[
   0.24 = 1.8d - 0.1d
   \]

   \[
   0.24 = 1.7d
   \]

   \[
   d = \frac{0.24}{1.7}
   \]

   \[
   d \approx 0.141 \, \text{meters} = 14.1 \, \text{cm}
   \]

Therefore, **the maximum distance between Nancy and the mirror for her to see the whole image of the man is approximately 14.1 cm**.

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Would you like further details on this solution, or do you have any specific questions?

Here are some additional questions for exploration:

1. What would happen if the mirror were longer? How would this affect the maximum distance \( d \)?
2. How would the scenario change if the man were standing closer to Nancy?
3. What is the minimum height Nancy needs to place the mirror at for her to see the entire height of the man?
4. If the man were taller, how would this affect the maximum distance \( d \) between Nancy and the mirror?
5. How does the height of Nancy's eyes affect the maximum distance \( d \)?

**Tip:** When using mirrors to see objects behind you, the distance between you and the mirror can significantly affect what you see. Understanding the geometry of reflection helps in optimizing mirror placement for the best view.

Calculate the maximum distance Nancy can be from a mirror to see the entire image of a man standing 2.4m behind her. Includes step-by-step solution using principles of geometry, optics, and similar triangles. Solve for mirror size and Nancy's viewing distance.

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