Mirror Size & Height: Solving the Puzzle

[Icheb]

There is a woman who is 1,75m tall standing in front of a mirror. Her eyes are to be 10cm below the top of her head. I have to find out how small the mirror can be so that she can still see her entire body in it and how high it has to be hanging on top of the floor.

Now I think I can solve it by drawing the problem. If I draw a line from the top of her head and from her feet of the virtual image to her eyes, the highest point of the mirror would have to be 5cm above her eyes and 1,65m/2 below her eyes. Assuming this is correct (which I'm not sure of), what equations would I use to write this down? Somehow the graphical solution is all I can come up with.

 

[Shooting Star]

You are quite right. Use similar triangles.

 

[ogb p]

I would approach this problem by considering the principles of optics and geometry. The key concept to consider is the angle of reflection, which states that the angle at which light reflects off a mirror is equal to the angle at which it strikes the mirror. In this case, the angle of reflection would need to be large enough for the woman to see her entire body in the mirror.

To solve for the minimum size of the mirror, we can use the equation: θ = tan^-1 (h/d), where θ is the angle of reflection, h is the height of the woman's eyes above the ground, and d is the distance from her eyes to the mirror. In order for the woman to see her entire body, the angle of reflection would need to be at least 90 degrees (assuming she is standing upright). Therefore, we can set θ = 90 degrees and solve for d to find the minimum distance from her eyes to the mirror.

To determine the height at which the mirror needs to be hung, we can use the equation: h' = h + (2d * tan θ), where h' is the height at which the mirror needs to be hung. Again, setting θ = 90 degrees, we can solve for h' to find the minimum height at which the mirror needs to be hung.

In summary, to find the minimum size of the mirror and the height at which it needs to be hung, we would use the equations θ = tan^-1 (h/d) and h' = h + (2d * tan θ) with the constraint that θ = 90 degrees. This approach takes into account the principles of optics and geometry to provide a mathematical solution to the problem.