A woman fixing her hair holds a plane mirror 30 cm away from the front of her face so as to look into a plane mirror on the bathroom wall behind her. She is 90 cm from the wall mirror. Approximately how far does the image of the back of her head appear in front of her?

240 cm

240cm.

The ray of light reflected from her head travels 90cm to the mirror on the bathroom wall, and the ray of light travels 120cm (90+30) back to the mirror in front of her before travelling into her eyes 30cm away from the mirror.
=>90 + 120 + 30 = 240cm.
Gracias.

210 because 2 pane mirrors are parallel to each other and we have to find the distance between the image of back head in the wall mirror and and the

infront mirror

+30 because they are asking from the object

To determine approximately how far the image of the back of the woman's head appears in front of her, we'll use the concept of image formation by plane mirrors.

First, let's visualize the situation. We have a woman standing in front of a plane mirror (mirror 1), holding another plane mirror (mirror 2) at a distance of 30 cm from her face. The wall mirror (mirror 3) is behind her at a distance of 90 cm.

Now, let's break down the process:

1. The woman's face reflects off mirror 2 (30 cm away) towards mirror 3 (90 cm away).
2. The reflection off mirror 3 travels back towards mirror 2.
3. The image of the back of the woman's head is formed where the reflection off mirror 3 intersects with mirror 2.

To find the distance that the image of the back of her head appears in front of her, we'll use the mirror formula:

1/f = 1/v + 1/u

Where:
- f is the focal length of the mirror (which is infinite for a plane mirror),
- v is the image distance,
- u is the object distance.

In this case, we are given the object distance (30 cm) and the image distance (90 cm).

Plugging these values into the mirror formula, we have:

1/f = 1/90 + 1/30

Simplifying the equation, we get:

1/f = 1/45 + 2/90

1/f = 3/90

1/f = 1/30

Therefore, the focal length (f) is 30 cm.

In the case of a plane mirror, the image distance (v) is equal to the object distance (u).

So, the image of the back of the woman's head appears in front of her at a distance of approximately 30 cm.

Therefore, the image of the back of her head appears approximately 30 cm in front of her.