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?
a man can see upper 4 ft of his height, i.e. 2/3 of his height in the both locations.
To answer these questions, let's break down the scenarios step by step:
1. How much of his image can the man see when he is standing in front of the mirror?
To determine this, we need to consider the geometry of the situation. The man's eyes are at the top of his head, which is 6 ft above the ground. The mirror is placed 4 ft above the ground. Therefore, the man can see his entire body from the ground up to his eye level, which is 6 ft - 4 ft = 2 ft of his image.
2. How much of the 8 ft-tall tree behind him can he see when he's in front of the mirror?
The man is standing 5 ft away from the wall, where the mirror is placed. The tree is 10 ft away from the wall. The mirror reflects the image of the tree, but since the mirror is only 2 ft in length, the man can only see a portion of the reflected tree. To determine this, we need to calculate the angle formed between the man's line of sight and the top of the tree.
Assuming the man is looking straight ahead, the angle formed can be found using trigonometry. We can use the tangent function, which is opposite over adjacent sides.
tan(θ) = opposite/adjacent = height of the tree/ distance from the wall to the mirror.
Using these values, we have:
tan(θ) = 8 ft / 10 ft = 0.8
To find the angle θ, we can use the inverse tangent function (arctan) on both sides:
θ = arctan(0.8) ≈ 38.66 degrees
Therefore, the man can see the portion of the tree that is within a 38.66-degree angle above the horizontal line of the mirror.
3. When the man moves to stand next to the tree, how much of himself can he now see?
Since the mirror stays in the same place, the angle at which the man can see himself remains the same. Therefore, the man can still see 2 ft of his image.
4. How much of the tree can he now see when standing next to it?
Now that the man is next to the tree, the distance between him and the tree is negligible, and we can assume that the angle at which he sees the tree is close to 90 degrees. Therefore, the man can see the entire height of the tree, which is 8 ft.