A man who is 6 ft tall is standing in front of a plane mirror that is 2 ft in length. If the mirror is placed lengthwise with its bottom edge 4 ft above the floor on a wall that is 5 ft away, how much of his image (i.e. what length of himself) can the man see? (Assume that his eyes are right at the top of his head). How much of the 8 ft-tall tree behind him, 10 ft away from the wall, can he see? When the man moves to stand next to the tree, with the mirror staying in place, how much of himself can he now see? How much of the tree can he now see?
To determine how much of the man's image he can see in the mirror, we need to consider the height of the man and the position of the mirror.
Given:
Height of the man = 6 ft
Length of the mirror = 2 ft
Distance of mirror's bottom edge from the floor = 4 ft
Distance of the mirror from the wall = 5 ft
To calculate the portion of the man's image visible in the mirror, we can consider the triangle formed by the man, the mirror, and the floor.
Using similar triangles, we can determine the height of the man's image in the mirror. The height of the man plus the height of his image in the mirror will be equal to the total distance from the bottom of the mirror to the top of the mirror.
Let's calculate the height of the man's image:
Height of the mirror = 4 ft (distance of mirror's bottom edge from the floor) + 2 ft (length of the mirror) = 6 ft
Now, let's calculate the height of the man's image in the mirror by subtracting the man's height from the height of the mirror:
Height of the man's image = Height of the mirror - Height of the man
= 6 ft - 6 ft
= 0 ft
This means that the man cannot see any part of his image in the mirror since his eyes are right at the top of his head.
Now, let's calculate how much of the 8 ft-tall tree behind him, which is 10 ft away from the wall, he can see.
Since the mirror is placed lengthwise, the man can see the reflection of the tree within the field of view not blocked by his own body.
To calculate this, we need to determine the field of view blocked by the man's body, and then subtract it from the total height of the tree.
Let's calculate the field of view blocked by the man's body:
Distance from the wall to the mirror = 5 ft
Distance from the man to the mirror = 5 ft
The field of view blocked by the man's body will form a right triangle with the man, the mirror, and the wall. The angle at the man's head will be the same as the angle made by the mirror, the man, and the tree behind him.
Using trigonometry, we can calculate the angle:
tan(theta) = opposite/adjacent = Height of the man / (Distance from the wall to the mirror + Distance from the man to the mirror)
tan(theta) = 6 ft / (5 ft + 5 ft) = 0.6
theta = arctan(0.6) ≈ 30.96 degrees
30.96 degrees is the angle at the man's head, which indicates the angle of the field of view blocked by his body.
Now, let's calculate the portion of the tree that the man can see:
Angle of view = 180 degrees - angle at the man's head = 180 degrees - 30.96 degrees ≈ 149.04 degrees
The man can see approximately 149.04 degrees of the tree, considering that the mirror fully reflects the tree within the man's field of view.
Now, let's consider when the man moves to stand next to the tree, with the mirror staying in place. We need to calculate how much of himself he can now see, and also how much of the tree he can see.
Since the man is now standing next to the tree, the mirror will be reflecting only the image of the tree, not the man.
The man can see his entire body directly, without relying on the mirror. Therefore, he can see his full height, which is 6 ft.
As for the tree, the mirror's position did not change, so the portion of the tree the man can see will remain the same as before - approximately 149.04 degrees.