A person stands 4.4 m in front of a wall that is covered floor-to-ceiling with a plane mirror. His eyes are 1.9 m above the floor. He holds a flashlight between his feet and manages to point it at the mirror. At what angle of incidence must the light strike the mirror so the light will reach his eyes?
The beam must strike the mirror half way up from his feet to his eyes. The angle is incidence is therefore
tan^-1 [(1/2)*1.9/4.4] = tan^-1 .21591
= 12.2 degrees
Well, if this person wants to see their reflection, they better not be a vampire! Let's get to the question. To find the angle of incidence, we'll need a little bit of trigonometry.
Let's draw the scenario. We have the person standing in front of the mirror, and we want to find the angle at which the light hits the mirror.
Now, we can draw a right triangle from the person's eye to the point where the light hits the mirror. The vertical leg of the triangle is the distance between the person's eyes and the floor (1.9 m), and the hypotenuse is the distance between the person and the mirror (4.4 m).
Using some trigonometry magic (sine function), we can find the angle. The sine of the angle of incidence is equal to the opposite side (1.9 m) divided by the hypotenuse (4.4 m). So, we can write:
sin(angle) = 1.9 m / 4.4 m
Now we simply take the inverse sine (sin^(-1)) of that ratio to find the angle. Plug in the numbers into a calculator, and you'll get your answer.
Just remember, this answer assumes that the person is not bending their light with a magic trick or a clown trick. Otherwise, all bets are off!
To find the angle of incidence at which the light must strike the mirror so that it reaches the person's eyes, we can use the law of reflection. According to this law, the angle of incidence is equal to the angle of reflection.
In this case, the person's eyes are 1.9 m above the floor, and they are standing 4.4 m in front of the mirror.
Step 1: Draw a diagram to visualize the situation.
Let's draw a horizontal line to represent the floor, and a vertical line to represent the mirror. Mark the person's position standing in front of the mirror, with the eyes 1.9 m above the floor level. Also, draw a dashed line from the person's eyes to the mirror, representing the path of the reflected light.
Step 2: Find the distances involved.
The distance from the person's eyes to the mirror is 1.9 m, and the distance from the person to the mirror is 4.4 m. Since the light travels from the person's eyes to the mirror and then back to the eyes, the total distance traveled by the light is twice the distance from the person to the mirror, which is 2 * 4.4 m = 8.8 m.
Step 3: Determine the height at which the light reaches the mirror.
Since the light travels at the same angle of incidence as the angle of reflection, we can draw a right triangle using the person's eyes, the mirror, and the point where the light hits the mirror. The height of the triangle will be the height at which the light reaches the mirror. Let's call this height 'h'.
Step 4: Use the properties of the right triangle to find the height 'h'.
The length of one side of the right triangle is 1.9 m (the distance from the eyes to the floor), and the length of the other side is 8.8 m (the total distance traveled by the light). Let's use the Pythagorean theorem to find the height 'h'.
(1.9 m)^2 + h^2 = (8.8 m)^2
3.61 m^2 + h^2 = 77.44 m^2
h^2 = 77.44 m^2 - 3.61 m^2
h^2 = 73.83 m^2
h = √73.83 m
h ≈ 8.59 m
Step 5: Find the angle of incidence.
To find the angle of incidence, we can use the tangent function, which is the opposite side (height 'h') divided by the adjacent side (distance from the person to the mirror of 4.4 m).
tanθ = h / 4.4 m
tanθ = 8.59 m / 4.4 m
θ = tan^(-1)(8.59 m / 4.4 m)
θ ≈ 63.3°
Therefore, the light must strike the mirror at an angle of incidence of approximately 63.3° so that it reaches the person's eyes.
To find the angle of incidence at which the light must strike the mirror so that it reaches his eyes, we can use the law of reflection. According to the law of reflection, the angle of incidence (θi) is equal to the angle of reflection (θr).
In this case, the person stands 4.4 m in front of the wall, and his eyes are 1.9 m above the floor. Since the mirror is floor-to-ceiling, the distance from the floor to the person's eyes is also 4.4 m.
To visualize this situation, imagine a right-angled triangle formed by the person, the floor, and the mirror. The hypotenuse of this triangle represents the distance from the person's eyes to the mirror, which is 4.4 m. The vertical leg of the triangle represents the height of the person's eyes above the floor, which is 1.9 m. The horizontal leg represents the distance from the person's eyes to the mirror along the floor, which is the same as the distance between the person and the mirror, 4.4 m.
Now, let's calculate the distance along the floor from the person's feet to the point where the flashlight strikes the mirror. Based on the given information, we know that the person's eyes are 1.9 m above the floor, and the distance from the person to the mirror is 4.4 m. Therefore, the distance along the floor from the person's feet to the mirror must be 4.4 m - 1.9 m = 2.5 m.
In the right-angled triangle, we have the opposite side (1.9 m) and the adjacent side (2.5 m). To find the angle of incidence, we can use the tangent function:
tan(θi) = opposite / adjacent
tan(θi) = 1.9 m / 2.5 m
Now, we need to inverse tangent (arctan) both sides of the equation to find the angle:
θi = arctan(1.9 m / 2.5 m)
Using a calculator, the angle of incidence (θi) can be calculated as approximately 39.81 degrees.
Therefore, the light must strike the mirror at an angle of approximately 39.81 degrees for it to reach the person's eyes.