You are 1.9 m tall and stand 3.3 m from a plane mirror that extends vertically upward from the floor. On the floor 1.6 m in front of the mirror is a small table, 0.75 m high.

What is the minimum height the mirror must have for you to be able to see the top of the table in the mirror?

To find the minimum height the mirror must have for you to be able to see the top of the table in the mirror, we need to consider the angle of incidence and the angle of reflection.

First, let's determine the angle at which you can see the top of the table. This angle is formed by a line drawn from your eyes to the top of the table and a line drawn from your eyes to the mirror.

Using trigonometry, we can find the angle using the following equation:

angle of incidence = arctan(table height / distance to table)

In this case, the table height is 0.75 m, and the distance to the table is 1.6 m. Plugging these values into the equation:

angle of incidence = arctan(0.75 / 1.6)

Calculating this, the angle of incidence is approximately 25.9 degrees.

The angle of reflection is equal to the angle of incidence, as per the law of reflection. So, you should be able to see the top of the table in the mirror when the light from the top of the table reflects off the mirror at an angle of approximately 25.9 degrees.

Now, let's consider the mirror height required. To see the top of the table in the mirror, the reflected light rays must enter your eyes. To achieve this, the reflected rays must be angled upwards toward your eyes. Therefore, the mirror needs to be at a sufficient height, such that the reflected rays can reach your eyes at the desired angle.

To find the minimum height of the mirror, we use the concept of similar triangles. The height of your eyes from the floor is 1.9 m, and the distance between you and the mirror is 3.3 m. Thus, the distance from your eyes to the mirror's base is given by:

distance from eyes to mirror base = (distance to mirror) - (height of your eyes)

Plugging in the values:

distance from eyes to mirror base = 3.3 m - 1.9 m
distance from eyes to mirror base = 1.4 m

Therefore, we need the top of the table to be at the same height in the mirror as your eyes are above the mirror's base. So, the minimum height the mirror must have is 1.4 m for you to be able to see the top of the table in the mirror.