Solve Reflection Equation: Find Height of Shortest Mirror for Viewing Full Image

[moonlit]

I have a homework problem that I'm not sure how to figure out. What equation should I use for this?

A person whose eyes are 1.8 m above the floor stands in front of a plane mirror. The top of her head is 0.12 m above her eyes. What is the height of the shortest mirror in which she can see her entire image?

 

[moonlit]

Haha, nevermind. Figured it myself, thanks anyway.

 

[M98Ranger]

To solve this reflection equation, you 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 the angle between the person's eyes and the mirror, and the angle of reflection is the angle between the person's eyes and their image in the mirror.

To find the height of the shortest mirror for viewing the full image, we can use the following equation:

h = d + 2h'

Where:
h is the height of the shortest mirror
d is the distance between the person's eyes and the mirror
h' is the height of the person's image in the mirror

We can rearrange this equation to solve for h':

h' = (h-d)/2

Now, we need to find the value of h'. To do this, we can use the law of reflection and set up the following equation:

tan θ = h'/d

Where:
θ is the angle of incidence, which is equal to the angle of reflection
h' is the height of the person's image in the mirror
d is the distance between the person's eyes and the mirror

Since we know that the angle of incidence and the angle of reflection are equal, we can substitute θ with the angle between the person's eyes and the mirror, which is equal to the angle between the person's eyes and their image in the mirror. This angle can be found using basic trigonometry:

tan θ = (h + 0.12)/d

Now, we can substitute this into our previous equation and solve for h':

h' = (h-d)/2 = (h + 0.12)/2

Finally, we can plug in the values given in the problem to find the height of the shortest mirror:

h' = (1.8 - 0.12)/2 = 0.84 m

Therefore, the height of the shortest mirror for viewing the full image is 0.84 m.