The Place you get your haircut has two nearly parallel mirrors 5.0m apart. As you sit in the chair, yoru head is 2.0 m from the nearer mirror. Looking toward this mirror, you first see your face and then, farther away, the back of your head. ( The mirrors need to be slightly nonparallel for you to be able to see the back of your head, but you can treat them as parallel in this problem.) How far away does the back of your head appear to be? Neglect the thickness of your head.

To find the distance to the back of your head, we have to consider the reflections of reflections between the two mirrors.

The distance from your head to the farther mirror will be 5 - 2 = 3 meters, since the mirrors are 5 meters apart and your head is 2 meters away from the nearer mirror.

Now, let's consider the reflection of your head from the farther mirror to the nearer mirror. It will seem like there's another head 3 meters behind the farther mirror. So this reflection "head" will appear to be at 3 + 3 = 6 meters away.

Next, let's consider the reflection of the reflected head from the nearer mirror to the farther mirror. This is essentially the image of the back of your head since the head is reflecting on both mirrors. To find the total distance to this reflection, we have to add the distance of the reflection head from the nearer mirror to the distance between the mirrors, which is: 6 + 5 = 11 meters.

So the back of your head appears to be 11 meters away when looking into the mirror.

To determine how far away the back of your head appears to be, we can use the concept of virtual images formed by mirrors.

The distance between the two nearly parallel mirrors is given as 5.0m. You are located at a distance of 2.0m from the nearer mirror. Let's call this mirror "Mirror A". The other mirror, which is farther away, we'll call "Mirror B".

Now, when you look into Mirror A, you see a virtual image of your face. This virtual image is formed by the reflection of light rays off the mirror, creating the illusion of an image behind the mirror.

Since the mirrors are nearly parallel, the light rays will bounce back at an angle to reach your eyes. This allows you to see the back of your head in the virtual image. The apparent distance between your head and its virtual image in the mirror will determine how far away the back of your head appears to be.

To solve for the distance between your head and its virtual image, we can use similar triangles formed by the mirror arrangement.

Let's consider the two triangles formed by your head, its virtual image, and the two mirrors:

Triangle 1: Formed by your head, the virtual image, and Mirror A.
Triangle 2: Formed by your head, the virtual image, and Mirror B.

The corresponding sides of these triangles are parallel, and since the mirrors are nearly parallel, triangles 1 and 2 are similar triangles.

Using the properties of similar triangles, we can set up the following ratio:

Distance of your head from Mirror A / Distance of virtual image from Mirror A = Distance of your head from Mirror B / Distance of virtual image from Mirror B

Plugging in the given values:
2.0m / 5.0m = Distance of your head from Mirror B / Distance of virtual image from Mirror B

Solving for the unknown, we get:
Distance of your head from Mirror B = (2.0m / 5.0m) * Distance of virtual image from Mirror B

Since the distance between the mirrors is 5.0m and you are 2.0m away from Mirror A, the remaining distance between you and Mirror B is (5.0m - 2.0m) = 3.0m.

Therefore, the distance of your head from Mirror B is:
(2.0m / 5.0m) * 3.0m = 1.2m

So, the back of your head appears to be 1.2 meters away when looking into the mirrors.

To determine how far away the back of your head appears to be, we can use the concept of virtual images formed by mirrors.

Given:
Distance between the mirrors (d) = 5.0 m
Distance from your head to the nearer mirror (d₁) = 2.0 m

Let's assume the distance from the back of your head to the second mirror (d₂) is the distance we need to find.

In this case, the light rays from the back of your head will reflect off the first mirror and then off the second mirror before reaching your eyes.

Using the concept of mirror reflection, we know that the light rays will bounce off the mirrors at the same angle at which they hit the mirrors. Therefore, the angle of incidence and the angle of reflection are equal.

Since the mirrors are nearly parallel, the angle of incidence θ will be very small. This means that we can approximate the tangent of the angle θ as the ratio of the vertical distance traveled (d₂) to the horizontal distance traveled (d).

So, we can write:

tan(θ) ≈ d₂ / d

Rearranging the formula, we get:

d₂ ≈ d * tan(θ)

Since θ is small, we can approximate tan(θ) as sin(θ). Therefore:

d₂ ≈ d * sin(θ)

Now, we need to find the value of sin(θ).

To do that, we can use the triangle formed by the first mirror, the second mirror, and the path of the light rays. In this triangle, the hypotenuse is d, the opposite side is d₂, and the adjacent side is d - d₁.

Using the sine function, we have:

sin(θ) = d₂ / d

Substituting the value of d₂ from the rearranged formula above, we get:

sin(θ) = (d * sin(θ)) / d

Simplifying, we find:

sin(θ) = sin(θ)

This equation shows that the angle θ is the same on both sides of the equation. Therefore, sin(θ) is equal to sin(θ).

Hence, the distance from the back of your head to the second mirror (d₂) is approximately 2.0m.