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, we can use the principles of geometry and optics. Let's break it down step by step:

1. How much of his image can the man see?
Since the mirror is placed lengthwise and the man is standing in front of it, his image will be reflected back towards him. The mirror is 2 ft in length, so the man will be able to see half of his own height. Therefore, he can see 3 ft of his image.

2. How much of the tree behind him can he see?
The tree is located 10 ft away from the wall, and the wall is 5 ft away from the mirror. Since the mirror is placed vertically (lengthwise), the reflected image will appear at the same distance behind the mirror as the object is in front of it. Therefore, the man and the tree are equidistant from the mirror horizontally.
Considering the height of the tree, which is 8 ft, and the distance between the man and the mirror, we can use similar triangles to find the height of the tree's image. The ratio of the height of the tree's image (T) to the distance to the mirror (D) will be equal to the ratio of the tree's actual height (8 ft) to the distance to the tree (10 ft):
T/D = 8/10
Simplifying this equation, we can find that the height of the tree's image is 6.4 ft. However, since the mirror is only 2 ft in length, the man can only see half of the tree's image that is within the mirror's height. Therefore, the man can see 3.2 ft of the tree.

3. When the man moves to stand next to the tree, how much of himself can he now see?
In this scenario, the man moves 5 ft closer to the mirror, keeping it in place. This means that he is now only 5 ft away from the mirror. Using similar triangles with the same logic as before, the man's image will have the same height-to-distance ratio as in the previous case (3 ft of image height / 5 ft of distance to the mirror).
Therefore, the man will now see 3 ft of his image, just as he did before.

4. How much of the tree can he now see?
Since the man and the tree are now the same distance away from the mirror (5 ft), the mirror will reflect the tree's image at the same height as before (6.4 ft), based on the similar triangles and ratios we established earlier. However, the mirror is still only 2 ft in height.
Therefore, the man can now see 2 ft of the tree's image, which is the height of the mirror.

To summarize:
- The man can see 3 ft of his own image when standing in front of the mirror.
- The man can see 3.2 ft of the 8 ft-tall tree when standing in front of the mirror.
- When the man moves to stand next to the tree, he can still see 3 ft of his own image.
- When the man moves to stand next to the tree, he can only see 2 ft of the tree's image.