How High Should the Mirror Be Placed for Full Reflection?

[Daniella_123x]

Homework Statement

A man stands 1.85m in front of a mirror hanging on a wall. His eyes are 10 cm below the top of his head. At what height from the floor, must the mirror be placed?

Homework Equations

Snells Law: n1*sin(omega)1=n2*sin(omega)2

The Attempt at a Solution

I converted everything to cm so 1.85= 185 cm
and 185-10 = 175 which is the height from his eyes downward
I just do not understand how to set it up where I could possible find the height of the mirror!

 

[Simon Bridge]

Welcome to PF;
The problem statement is incomplete - how would the man know if the mirror was the right height or not?
"At what height from the floor, must the mirror be placed"... to do what?

i.e. Usually the idea is that the Man has to be able to see his entire body in the mirror - in which case we also need to know the height of the mirror. If it is a full-length mirror, then it can sit on the floor. Could you work out the minimum size of mirror that would let the man see his entire body at that distance?

 

[Daniella_123x]

Well Question a) was: What is the shortest mirror which without being moved, will permit the man to see his entire body? I solved this knowing that to see a full height of a person, he needs a mirror which has a length of half his height, which is 185/2 = 92.5 But I don't understand what does this have to do the question b)? That all they give me

 

[Simon Bridge]

Well - if the mirror was sitting on the floor - would he be able to see his whole body? Or just his feet?
What about if the bottom of the mirror were level with his eyes?

 

[Nickie]

To solve this problem, we can use the fact that the angle of incidence is equal to the angle of reflection for a mirror. This means that the angle between the man's eyes and the mirror is the same as the angle between the mirror and the floor. We can use this information to set up a right triangle with the man's height (185 cm) as the hypotenuse, the distance from his eyes to the mirror (175 cm) as the opposite side, and the distance from the mirror to the floor (x cm) as the adjacent side.

Using the trigonometric identity, sin(theta) = opposite/hypotenuse, we can write the following equation:

sin(theta) = 175/185

We can then solve for theta by taking the inverse sine of both sides:

theta = sin^-1(175/185) = 54.3 degrees

Now, we know that the angle between the mirror and the floor is also 54.3 degrees. We can use this information to set up another right triangle, this time with the distance from the man's eyes to the mirror (175 cm) as the hypotenuse, the height of the mirror (h cm) as the opposite side, and the distance from the mirror to the floor (x cm) as the adjacent side.

Using the trigonometric identity, sin(theta) = opposite/hypotenuse, we can write the following equation:

sin(54.3) = h/175

We can then solve for h by multiplying both sides by 175:

h = 175*sin(54.3) = 142.6 cm

Therefore, the mirror must be placed at a height of 142.6 cm from the floor in order for the man to see his full reflection.