15.calculate the angle af refraction as for the given information in the question.40.574.644.6

To calculate the angle of refraction, we can use Snell's Law, which states that:

$n_1 \sin(\theta_1) = n_2 \sin(\theta_2)$

Given Information:

• Let's assume the provided values represent:
  • $n_1 = 1.0$ (refractive index of air)
  • $n_2 = 1.33$ (refractive index of water, or another medium)
  • $\theta_1$ (angle of incidence) is given as $40.5^\circ$ (the average of 40.574 and 40.644).

Steps to Solve:

1. Break Down the Problem
   Identify the variables:

   • $\theta_1 = 40.5^\circ$
   • $n_1 = 1.0$ (for air)
   • $n_2 = 1.33$ (for water or the second medium)

2. Relevant Concepts
   Use Snell's Law to relate the angles and the refractive indices.

3. Analysis and Detail
   Rearranging Snell's Law for $\theta_2$: $\sin(\theta_2) = \frac{n_1 \sin(\theta_1)}{n_2}$ Substituting the known values: $\sin(\theta_2) = \frac{1.0 \cdot \sin(40.5^\circ)}{1.33}$

4. Calculating the Sine Value
   First, calculate $\sin(40.5^\circ)$: $\sin(40.5^\circ) \approx 0.6561$ Now, substituting this value into the equation: $\sin(\theta_2) = \frac{0.6561}{1.33} \approx 0.493$

5. Determine $\theta_2$
   Finally, find $\theta_2$ using the arcsine function: $\theta_2 = \arcsin(0.493) \approx 29.5^\circ$

Final Answer

The angle of refraction $\theta_2$ is approximately $29.5^\circ$.