How Tall is the Tree in the Plane Mirror Problem?

[CaneAA]

Homework Statement

A woman stands between a vertical mirror, 0.5 meter tall, and a distant tree whose height is H. she is 1.0 m from the mirror and the tree is 9.0 m from the mirror. If she sees the tree just fill the mirror, what is H?

Homework Equations

m = (hi/ho)/(-di/do)

hi = image height = 0.5
ho = object height = H
di = image height
do = object height

The Attempt at a Solution

I'm trying to solve the problem using the magnification equation. The image height has to be the height of the mirror and the object height is the unknown.

However, I'm having a hard time figuring out what values to use for di and do. In all the problems I've done so far, "do" was the distance from the object to the mirror and "di" was the distance from the image to the mirror (equal to "do" but negative for a virtual image), but in this problem, I don't know how to incorporate the person's distance from the mirror.

The correct answer is 5 m.

 

[kuruman]

It looks like this is a plane mirror. If it is a curved mirror, you can't do the problem without the focal length or curvature. The magnification is +1, if you must have it. Draw yourself a neat geometric diagram and you will see what is going on.

 

[CaneAA]

I did draw a diagram--I just don't know what "do" or "di" would be in this case, since I can't figure out how to incorporate the person's distance from the mirror. I'm guessing that the person must factor into the "do" or "di" measurement somehow.

And I don't see how the magnification would help since 1 = 0.5/H would be H=0.5 and that is incorrect.

 

[kuruman]

There are two "objects" in this case, a woman and a tree, therefore there are two values for do, namely do1 = 1 m and do2 = 9 m. Draw a plane mirror of some height that you label h0 (0.5 m), then draw a woman in front of the mirror and a tree also in front of the mirror but at about nine times the woman's distance from the mirror. Label the tree's height H.

Question 1: What are di1 and di2? Draw the images in your figure.
Question 2: What condition must be met so that the woman sees the tree just fill the mirror? What does your diagram say?

 

[rwisz]

Hello,

Thank you for sharing the homework problem with me. It seems like you are on the right track by using the magnification equation to solve this problem. However, as you mentioned, the distances in this problem are a bit different than what you have encountered before.

In this case, di would be the distance from the person to the mirror (1.0 m) and do would be the distance from the mirror to the tree (9.0 m). This is because the image of the tree is being formed on the mirror, and the person is standing in front of the mirror at a distance of 1.0 m.

Using the magnification equation, we can set up the following equation:

m = (hi/ho)/(-di/do)

We know that hi = 0.5 m, di = 1.0 m, and do = 9.0 m. We can rearrange the equation to solve for ho, the height of the tree:

ho = (hi x do x m)/(-di)

Substituting in the values, we get:

ho = (0.5 m x 9.0 m x 1)/(-1.0 m) = 4.5 m

This is the height of the tree as seen in the mirror. However, since the tree is filling the mirror, the height of the tree in real life would be double that, giving us the final answer of 9.0 m.

I hope this explanation helps you understand the problem better. Keep up the good work with your studies!

Best,