Question

If the separation between the two eyes of a man is d, what should be the minimum width of the mirror in which the man could see the full image of his face of width w, using both of his eyes.

• A. $\frac{1}{2}(w+d)$
• B. $(w-d)$
• C. $\frac{1}{2}(w-d)$
• D. $\frac{1}{2}w$

Answer 1

The correct option is C $\frac{1}{2}(w-d)$

Lets draw the extreme rays from the face (We are using the top view to describe the solution).
There are two cases possible as mentioned in the diagrams.

Which of the cases will give us a minimum width of mirror ?
To pick one of those we should understand that the minimum width of the mirror will be given by the case in which the reflected rays will have to get deviated more to reach the eyes. (For a ray to reach the closest eye won't give you the minimum width of the mirror.)

So, now in the diagram,
We have,
$2x=d+\frac{(w-d)}{2}=\frac{w}{2}+\frac{d}{2}s=\frac{w}{4}+\frac{d}{4}$
We need y, from the diagram we have
$y=\frac{(w-d)}{2}-xWegety=\frac{w}{4}-\frac{3d}{4}$
Now, total width of the mirror = 2y+d
$=\frac{w}{2}-\frac{3d}{2}+d=\frac{w}{2}-\frac{d}{2}=\frac{1}{2}(w-d).$