Question

A person is 1.9 m tall. Where should he place the top of a mirror on the wall so he can see the top of his head? Assume his eyes are 5.4 cm below the top of his head. Find a minimal vertical distance from the level of the person’s eyes to the top of the mirror.

Answer 1

Answer: a) 1.846m

b) 0.923m

Explanation: given that height h = 1.9m

Assume his eyes are 5.4 cm below the top of his head

5.4 cm = 5.4 / 100 = 0.054 m

1.9 – 0.054 = 1.846 meter

The top of the mirror should be placed at 1.846m

That is the vertical distance from his eyes to top of his head.

Light must travel from the top of his head to the mirror in order for him to see the head. Then travel from this position to his eyes. To do this, the mirror must be placed at a position that is halfway between his eyes and the top of his head. 

The minimal vertical distance from the level of the person’s eyes to the top of the mirror will be:

d = ½ ×1.846 = 0.923 meters

Answer 2

Answer:

A) Height he should place the top of a mirror on the wall so he can see the top of his head is = 2.7cm above his eyes.

B) Minimal vertical distance from the level of the person’s eyes to the top of the mirror = 2.7cm

Explanation:

Height of eye below top of head = 5.4cm or 0.054m

Height of person = 1.9m

A) If we assume that the top of his head is (approximately) the same distance from the mirror as his eyes considering that the mirror/wall are vertical, then the top of the mirror for him to see the top of his head must be 0.054/2 = 0.027m or 2.7cm above the top of his eye.

B) The mirror must be at least half as tall as the person standing in front of it.

Since the person is 1.9m or 190 cm tall, the vertical dimension of the mirror must be at least 190/2 = 95 cm. Now, the lower edge of the mirror must be at a height that is half the distance between his feet and eyes. Since the eyes are located 5.4 cm from the top of his head, it means the eyes are at a height of 190 - 5.4cm = 184.6 cm from his feet.

Thus, the lower edge of the mirror must be = 184.6/2 = 92.3 cm from the ground.

The point of incidence of the ray from the 'top of her head' must lie at the midpoint between her eyes and the top of her head. Since the eyes are 5.4 cm below the top of her head, the point of incidence of the ray must be 190 cm - (5.4/2) cm = 187.3 cm from the ground. This marks the upper edge of the mirror.

Eyes are 184.6 cm from the ground, thus, minimal distance from eyes to top of mirror = 187.3 - 184.6 = 2.7 cm