It is necessary to illuminate the bottom of a well by reflected solar beam when the light is incident at an angle of `alpha = 40^(@)` to the vertical. At what angle `beta` to the horizontal should a plane mirror be placed ?

Options

A. `70^(@)`
B. `20^(@)`
C. `50^(@)`
D. `40^(@)`

Answer

To solve the problem of determining the angle \( \beta \) at which a plane mirror should be placed to illuminate the bottom of a well with a solar beam incident at an angle of \( \alpha = 40^\circ \) to the vertical, we can follow these steps:

### Step 1: Understand the Incident Angle
The angle of incidence \( \alpha \) is given as \( 40^\circ \) to the vertical. Therefore, the angle of incidence \( i \) with respect to the normal (which is perpendicular to the surface of the mirror) can be calculated as:
\[ i = 90^\circ - \alpha = 90^\circ - 40^\circ = 50^\circ \]

### Step 2: Apply the Law of Reflection
According to the law of reflection, the angle of incidence \( i \) is equal to the angle of reflection \( r \). Thus, we have:
\[ r = i = 50^\circ \]

### Step 3: Determine the Angle Between the Mirror and the Normal
The angle between the mirror and the normal can be found by recognizing that the angles of incidence and reflection are measured from the normal. Therefore, if the angle of incidence is \( 50^\circ \), the angle between the mirror and the normal is:
\[ \text{Angle between mirror and normal} = 90^\circ - i = 90^\circ - 50^\circ = 40^\circ \]

### Step 4: Calculate the Angle of the Mirror with Respect to the Horizontal
The angle \( \beta \) that the mirror makes with the horizontal can be calculated as follows:
- The angle between the vertical and the mirror is \( 40^\circ \) (from Step 3).
- Therefore, the angle \( \beta \) with respect to the horizontal is:
\[ \beta = 90^\circ - 40^\circ = 50^\circ \]

### Final Answer
Thus, the angle \( \beta \) at which the plane mirror should be placed with respect to the horizontal is:
\[ \beta = 50^\circ \]

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