A man 2m tall whose eye level is 1.84m above the ground looks at his image in a vertical mirror. If the man is to be above to see the whole of his body, determine the minimum length of the vertical mirror
To determine the minimum length of the vertical mirror, we need to consider the path of light from the man to the mirror and then back to his eyes.
Let's visualize the situation:
```
┌───────────────────┐
┌─────┼──────┐ │
│Man │ │ │Mirror│
│ │ └────┘ │
│ │ ┌─────┴────┐
├─────┤ │Man's Eye │
│Floor│ │ Level │
└─────┘ │ │
└──────────┘
```
We know that the man is 2 meters tall, and his eye level is at a height of 1.84 meters. Since the mirror is vertical, the light from his feet will reflect at the same angle as it strikes the mirror, and reach his eyes.
To see his whole body, the reflected light from his feet must reach his line of sight. This means that the reflected light must travel from his feet, hit the mirror, and reach his eyes.
Using the principle of reflection, we can determine the path of the light:
```
┌─────────────────────────────────┐
Pi Pf
│ /
│θ /
│ /
│/
Mirror
```
Here, Pi represents the initial path of light from the man's feet, Pf represents the final path of reflected light from the mirror, and θ represents the angle of incidence and reflection.
We can use the law of reflection, which states that the angle of incidence is equal to the angle of reflection. In this case, the angle of incidence is θ, and the angle of reflection is also θ.
Now, we can use basic trigonometry to find the relationship between the height of the man, his eye level, and the minimum length of the vertical mirror.
Starting with the right-angled triangle formed by the height of the man (2m), the height from the floor to his eye level (1.84m), and the distance from his feet to the mirror, we have:
```
┌─────┐
│ │
│ │
│θ │
└─────┘ 1.84m
2m
```
Using the tangent function:
tan(θ) = opposite/adjacent = 1.84m/x
We want to find the minimum length of the mirror, so we want to know the distance x.
Rearranging the equation to solve for x:
x = 1.84m / tan(θ)
To find the value of θ, we know that the line joining the man's eye level to his feet is perpendicular to the mirror surface. So, the angle between the line connecting the man's eyes to his feet and the line connecting the man's eyes to the mirror will be the complement of θ.
Therefore:
θ = 90° - arctan(1.84m/2m)
To calculate the value of θ in degrees (°), we can use a calculator.
Once we have the value of θ, we can substitute it back into the equation to find the minimum length of the vertical mirror:
x = 1.84m / tan(θ)
Calculating the value of x will give us the minimum length of the vertical mirror that allows the man to see his whole body.