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Practical Math - physics - not homework

by Ann
(Atlantic, Canada)











































Physics of Light

I need a mirror hung but I need to know how high a 48=inch high mirror must be mounted in order that someone six feet tall can see their entire image from no further back from the mirror than six feet. I'm certain there is a formula for this but alas I'm light years away from my school days so just don't remember. Help certainly would be appreciated--and thanks.

Comments for Practical Math - physics - not homework

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Dec 09, 2012
Physics of Light
by: Staff



Answer

Part I


How high should you hang the mirror?

You have a 1 foot leeway in how high you can hang your mirror.

I assume your mirror will be hung vertically (flat against the wall), and not tilted downward.

To meet your criteria, the top edge of the 48 in mirror can be hung anywhere between 6 and 7 feet above the floor.

In other words, the 48 inch mirror must be hung so that the top edge of the mirror is greater than 6 feet above the floor and less than 7 feet above the floor.

If the mirror is hung so that the top edge is below 6 feet, you will not be able to see the top of your head.

If the mirror is hung so that the top edge is above 7 feet, you will not be able to see your feet.



How far away from the mirror do you need to stand to see your entire image?

Your distance away from the mirror does not matter.

Whether you stand 1 foot away from the mirror, or 5 feet away from the mirror, you will be able to see your entire image (provided the top edge of the mirror is hung between 6 to 7 feet above the floor).

The image you will see when you stand 5 feet away from the mirror will be smaller than the image you see when standing 1 foot from the mirror, but you will see all of it.

The image you will see when you stand 1 foot away from the mirror will be large, but the entire image will be visible. At a distance of 1 foot it may strain you eyes to look down the mirror in order to see your feet.



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Dec 09, 2012
Physics of Light
by: Staff


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Part II


The Explanation

You can solve this problem by applying the Law of Reflection..

Imagine a line perpendicular to the surface of the mirror. This is the green “Normal” line shown in the following figure.


Math – perpendicular to surface of mirror







Now imagine a ray of light (the Incident Ray) striking and being reflected at the base of the green Normal line segment. The Incident Ray is shown in purple, and the Reflected Ray is shown in blue on the diagram shown below.

The Angle the Incident Ray makes with the Normal line segment is called the angle of incidence, θᵢ.

The Angle the Reflected Ray makes with the Normal line segment is called the angle of reflection, θᵣ.





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Dec 09, 2012
Physics of Light
by: Staff


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Part III





Math – light reflection from mirror:  angle of incidence, θ, and angle of reflection, θᵣ.





According to the “Law of Reflection”, the angle of incidence, θᵢ, will always be equal to the angle of reflection, θᵣ.

θᵢ = θᵣ

Since both angles are equal, they can both be represented by θ.


 Math – light reflection from mirror:  angle of incidence and angle of reflection are equal.








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Dec 09, 2012
Physics of Light
by: Staff


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Part IV

If a person is standing in front of your mirror, the light from that person’s shoes must be reflected from the mirror at an angle which will reach the person’s eyes . . . and, the light from the top of the person’s head must also be reflected at an angle which will reach the person’s eyes.

 Math – Since the angle of incidence and the angle of reflection must be equal, the light from a person’s shoes must be reflected in the mirror exactly ½ of the distance between the person’s eyes and the person’s shoes.








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Dec 09, 2012
Physics of Light
by: Staff


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Part V


If the angle, θtop of head, is too great, the person will not be able to see his head.

If the angle, θshoes, is too great, the person will not be able to see his or her shoes.

Since the angle of incidence and the angle of reflection must be equal, the light from a person’s shoes must be reflected in the mirror exactly ½ of the distance between the person’s eyes and the person’s shoes.

(In order to simplify the calculation, I’m going to assume a person’s eyes are at the top of his or her head, rather than 2 or 3 inches lower.)

If the person looking in the mirror is 6 feet tall, the light from a person’s shoes must always be reflected in the mirror at exactly 3 feet above the floor (half of the person’s height).

This will be true no matter how far away from the mirror the person is standing.

The lower edge of the mirror must always be less than 3 feet above the floor. If it is higher than 3 feet, a 6 foot tall person will not be able to see their feet.

Since the bottom edge of the 48 inch mirror must be less than 3 feet above the floor, the top edge of the mirror must be less than 7 feet above the floor (48 inches + 3 feet = 7 feet).






Thanks for writing.

Staff
www.solving-math-problems.com



Dec 09, 2012
Thanks so much
by: Ann

You have no idea how appreciative I am for your answers. I'm an oldster and really do wish I'd have paid more attention to math as a student. Guess I was just too immature way back then to realize what astounding applications it has. Besides, I was too easily intimated by it all. Let that be a lesson for anyone here doing homework--don't ever let the subject of math get away from you.

This truly is a fabulous site for those who are willing to seek help with the universal language of math. And for those of you who are involved with helping students,you are an amazing group of dedicated persons who deserve all manner of praise.

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