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 answer these questions, let's break down the steps and understand the concepts involved.
1. Determining the length of the man's image:
To find out how much of his image the man can see, we first need to consider the distance between the man and the mirror and the height of the mirror. The man is 6 ft tall, and the mirror is 2 ft tall. Since the mirror is lengthwise, the whole 6 ft height of the man will be visible. Therefore, the man can see his entire image in the mirror.
2. Determining the portion of the tree visible from the man's position:
Since the tree is behind the man and the mirror is flat, the mirror will reflect an image of the tree. To calculate how much of the tree is visible in the mirror, we need to visualize the alignment. The man is standing 4 ft in front of the mirror, and the tree is 10 ft away from the wall.
Using similar triangles, we can calculate the height of the reflected image of the tree in the mirror. The distance between the man and the mirror (4 ft) is related to the distance between the mirror and the tree (10 ft) as the height of the man (6 ft) is related to the height of the tree's reflection. Setting up a proportion:
(Height of tree reflection) / (Height of man) = (Distance between mirror and tree) / (Distance between man and mirror)
Let's solve for the height of the tree's reflection:
(Height of tree reflection) / 6 ft = 10 ft / 4 ft
Cross-multiplying, we get:
(Height of tree reflection) = (6 ft * 10 ft) / 4 ft
Height of tree reflection = 15 ft
So, the man can see 15 ft of the tree's reflection in the mirror.
3. When the man moves next to the tree:
Now, let's consider when the man moves next to the tree while the mirror remains in place. With the mirror still being 2 ft tall, the man's image in the mirror will remain the full 6 ft. However, we need to calculate how much of his body is visible outside the mirror.
Since the mirror is still 2 ft tall, only the portion of the man's body above the mirror will be visible. The man's height is 6 ft, and the mirror is 2 ft tall. Therefore, the man can see 4 ft of his image outside the mirror (6 ft - 2 ft).
4. Determining the portion of the tree visible from the new position:
Now that the man is standing next to the tree, his view of the tree is unaffected by the mirror. So, he can see the entire height of the tree, which is 8 ft.
To summarize:
- The man can see his full 6 ft image in the mirror.
- Initially, the man can see 15 ft of the tree's reflection in the mirror.
- When the man moves next to the tree, he can see 4 ft of himself outside the mirror, and he can see the full 8 ft height of the tree.