To determine the angle at which a light beam must be directed at the surface of the water to illuminate a coin lying at the bottom of the pool, we need to apply some principles of trigonometry.
Step-by-Step Solution:
1. Identify the Problem:
- Depth of water ([tex]\( d_w \)[/tex]) = 1.5 meters
- Horizontal distance from the wall to the coin ([tex]\( d_h \)[/tex]) = 0.90 meters
2. Understand the Setup:
- The light beam needs to enter the water at an angle. We need to calculate this angle of incidence.
- When the light travels from one medium (air) to another (water), its path bends due to refraction. However, because we are looking at the situation on the same plane and for simplicity, we assume the incident angle needed is the same as the refracted one.
3. Apply Trigonometry:
- Consider the triangle formed by the side wall, the vertical depth of the pool, and the horizontal distance to the coin.
- We need to find the angle [tex]\( \theta \)[/tex] at which the light must enter the water.
- Using right angle trigonometry, we find the tangent of [tex]\( \theta \)[/tex]:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{d_w}{d_h}
\][/tex]
Where:
- [tex]\( \text{opposite} \)[/tex] is the depth of the water (1.5 meters)
- [tex]\( \text{adjacent} \)[/tex] is the distance from the wall to the coin (0.90 meters)
4. Calculate the Angle:
- Compute [tex]\( \theta \)[/tex] using the arctangent (inverse tangent) function.
[tex]\[
\theta = \arctan\left( \frac{1.5}{0.90} \right)
\][/tex]
- The result of this calculation in radians and converting it to degrees:
[tex]\[
\theta \approx 1.0304 \text{ radians}
\][/tex]
[tex]\[
\theta \approx 59.0362^\circ
\][/tex]
Conclusion:
For the light beam to illuminate the coin at the bottom of the pool, it must be directed at an angle of approximately [tex]\( 59.0362 \)[/tex] degrees to the surface of the water.
