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Chapter 9: Problem 14

Compared to a person's height, what is the minimum length (top to bottom) of a mirror that will allow the person to see a complete image from head to toe?

Short Answer

Expert verified
Answer: The minimum length of the mirror required is half the height of the person.

Step by step solution

01

Identify the problem and visual representation

We need to find the minimum length of a mirror that allows a person to see their complete image from head to toe. To visualize the problem, imagine a person standing in front of a mirror on a flat surface, with the mirror being vertical and perpendicular to the floor.
02

Use the law of reflection

We'll analyze this problem by considering the law of reflection. The law of reflection states that the angle of incidence (the angle between the incoming light ray and the normal) is equal to the angle of reflection (the angle between the reflected light ray and the normal). The normal is an imaginary line perpendicular to the surface of the mirror.
03

Person's view of their head

In order for the person to see their head, a ray of light must bounce off the top of their head, hit the mirror, and then be reflected into their eyes. Because the angle of incidence is equal to the angle of reflection, the person's eyes must be positioned at the same height from the floor as the position where the light ray from the top of their head strikes the mirror.
04

Person's view of their feet

In order for the person to see their feet, a ray of light must bounce off their feet, hit the mirror, and then be reflected into their eyes. Again, because the angle of incidence is equal to the angle of reflection, the position where the light ray from the feet strikes the mirror must also be at the same height from the floor as the person's eyes.
05

Calculate the required mirror length

Since we determined that the mirror must reflect light from both the person's head and feet into their eyes, and considering that the person's eyes are positioned at half of their total height (assuming the eyes are at the midpoint of the person's height), we can now conclude that the mirror must be at least half the height of the person to reflect the complete image of the person from head to toe. Therefore, the minimum length (top to bottom) of the mirror is: Minimum Mirror Length = \frac{1}{2} \times Person's Height

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