Geometric Optics problem

[blumfeld0]

Homework Statement

If a plane mirror is hung at an angle of 30 degrees from the vertical toward the viewer. what is the minimum length that will allow the viewer to see the full body image if the line of vision can not be depressed below the horizontal. you may assume that the eyes are at the top of your head.

Homework Equations

N/A

The Attempt at a Solution

I have done a very similar problem where I know the height of the person. in that problem we did NOT assume that the eyes were on top of the head.
then the minimum length i need for the PLANE FLAT mirror (not tilted at an angle of 30 degrees) is HALF the height of the person.
so if the height of the person was 2 ft then the minimum length we need would be 1 ft for the mirror to see the full image of the person.

so my question is in this problem the mirror is tilted by 30 degrees. how does affect anything??
i do not think it does but i can not prove. i THINK the answer should still be half the height of object (person) but i might be wrong.

any ideas or solution so i can understand.

thank you

 

[blumfeld0]

After taking to someone they said that the 30 degrees will make a difference? That is counter to what I thought.
So I am guessing that the minimum length of the mirror will be

L = (2/Sqrt[3]) * height of person

I used Cos[30]= height/Length of mirror

is that even close?
if not any suggestions would be greatly appreciated

 

[Moetasim]

I would approach this problem by using the principles of geometric optics. First, let's define the variables in this problem: h represents the height of the person, d represents the distance between the person and the mirror, and L represents the minimum length of the mirror needed for the viewer to see the full body image.

To start, we can use the law of reflection which states that the angle of incidence is equal to the angle of reflection. In this case, the angle of incidence is 30 degrees, so the angle of reflection is also 30 degrees. This means that the reflected ray from the top of the person's head will hit the mirror at a 30 degree angle.

Next, we can use the concept of similar triangles. The triangle formed by the person, their image in the mirror, and the point where the reflected ray hits the mirror is similar to the triangle formed by the person, the person's reflection in the mirror, and the point where the person's reflection hits the mirror. This means that we can set up the following proportion:

h/d = h/L

Solving for L, we get L = dh/h. Since the angle of incidence is 30 degrees, we can use trigonometry to find the value of d in terms of h. Using the tangent function, we get d = h/tan 30 degrees = h/√3. Substituting this into our equation for L, we get L = h/√3 * h/h = h/√3.

Therefore, the minimum length of the mirror needed for the viewer to see the full body image is h/√3, which is approximately 0.577h. This means that the minimum length of the mirror is slightly more than half the height of the person. This is because the mirror is tilted at an angle, so the reflected ray will not hit the mirror at a 90 degree angle, resulting in a longer distance needed for the reflected ray to reach the viewer's eyes.

I hope this explanation helps you understand why the angle of the mirror does affect the minimum length needed for the viewer to see the full body image. It is important to consider all the variables and principles of geometric optics in order to solve this problem accurately.