Answer:
Step-by-step explanation:
A) The apparent depth of the fish when viewed at normal incidence to the water can be calculated using Snell's law:
n1 * sin(θ1) = n2 * sin(θ2)
where n1 and n2 are the indices of refraction of the first and second media, respectively, and θ1 and θ2 are the angles of incidence and refraction, respectively.
Since the fish is floating motionless, the angle of incidence is zero.
n1 = 1 (air)
n2 = 1.33 (water)
θ1 = 0
Solving for θ2, we have:
sin(θ2) = (n1/n2) * sin(θ1) = (1/1.33) * sin(0) = 0
Since sin(θ2) = 0, θ2 = 0, meaning that the light passes through the water without refraction. The apparent depth of the fish is simply its actual depth, 7.10 cm.
B) The apparent depth of the reflection of the fish in the bottom of the tank can be calculated using the same method as above, but with θ1 equal to the angle of reflection instead of incidence.
Since the bottom of the tank is a mirror, the angle of incidence is equal to the angle of reflection, which we can call θ.
θ1 = θ
Solving for θ2, we have:
sin(θ2) = (n1/n2) * sin(θ1) = (1/1.33) * sin(θ) = sin(θ) / 1.33
Next, we use the fact that the angle of incidence and the angle of refraction are related by the law of reflection:
θ1 = θ2
Setting these two equations equal to each other and solving for sin(θ), we find:
sin(θ) = 1.33 * sin(θ)
sin(θ) = 0.75
Taking the arcsine of both sides, we find:
θ = sin^-1(0.75) = 53.13 degrees
Finally, using the fact that the apparent depth is equal to the actual depth divided by the cosine of the angle of incidence, we find:
apparent depth = 7.10 cm / cos(θ) = 7.10 cm / cos(53.13 degrees) = 19.4 cm.