Hanging a Mirror: Find Optimal Size for Dressing Room

[sarahefried]

Homework Statement

You are preparing to go shopping for a plane mirror that will be hung on the wall of your dressing room. Your overall height is H = 2.4 m, and the vertical distance from the floor to the mid-line of your eyes is H1 = 2.23 m. You want the mirror to be tall enough to allow you to view your entire body from the tip of your head to the soles of your feet, while standing a horizontal distance D = 1.1 m from the mirror. Being frugal, you want to purchase the smallest possible mirror.

Homework Equations

There is only one such ray that obeys the law of reflection: angle of incidence equals angle of reflection: θi = θr

The Attempt at a Solution

I have absolutely no clue where to start!

 

[tiny-tim]

Welcome to PF!

You are preparing to go shopping for a plane mirror that will be hung on the wall of your dressing room. Your overall height is H = 2.4 m, and the vertical distance from the floor to the mid-line of your eyes is H1 = 2.23 m. You want the mirror to be tall enough to allow you to view your entire body from the tip of your head to the soles of your feet, while standing a horizontal distance D = 1.1 m from the mirror. Being frugal, you want to purchase the smallest possible mirror.

Welcome to PF!

Hint: draw the image of yourself on the other side of the mirror.

How large does the mirror have to be so that you can see the whole image without moving your head?

 

[thomate1]

I would approach this problem by first understanding the purpose of the mirror in the dressing room. The goal is to have a mirror that is both functional and cost-effective.

To determine the optimal size of the mirror, we need to consider two factors: the viewing angle and the cost.

The viewing angle is determined by the height and distance of the viewer from the mirror. In this case, the viewer's height is 2.4 m and the distance from the mirror is 1.1 m. Using the law of reflection, we can calculate the minimum height of the mirror needed to see the entire body. This would be the height of the viewer's eyes (2.23 m) plus the distance from the viewer to the mirror (1.1 m), giving us a minimum height of 3.33 m.

However, this is not a practical solution as it would require a very large and expensive mirror. Therefore, we need to find a balance between functionality and cost.

One option would be to purchase a mirror that is slightly taller than the minimum height calculated, such as 3.5 m. This would still allow the viewer to see their entire body while standing at a distance of 1.1 m from the mirror.

Another option would be to purchase a mirror that is slightly wider, rather than taller. This would allow the viewer to see their entire body by moving closer to the mirror, without the need for a very tall mirror.

Ultimately, the optimal size of the mirror will depend on personal preference and budget. I would recommend experimenting with different sizes and distances to find the most cost-effective and functional solution for the dressing room.