below the surface of still water. A boat is located where an incident ray from the fish encounters the boundary, and its refracted ray runs parallel to the surface. Find the horizontal distance r from the point on the surface directly above the fish to the boat.
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To solve this problem, we can use Snell's Law, which relates the angles of incidence and refraction when light passes from one medium to another. Snell's Law states:
n1 * sin(theta1) = n2 * sin(theta2)
where:
- n1 and n2 are the indices of refraction of the initial and final mediums, respectively
- theta1 and theta2 are the angles of incidence and refraction, respectively
In this case, the initial medium is air and the final medium is water, so we have:
n1 * sin(theta1) = n2 * sin(theta2)
1 * sin(theta1) = 1.33 * sin(90 degrees - theta2)
Since the boat is located where the incident ray encounters the boundary, it means that theta1 is the angle of incidence, and since the refracted ray runs parallel to the surface, it means theta2 is 0 degrees. Therefore, we can rewrite Snell's Law as:
sin(theta1) = 1.33 * sin(90 degrees)
Now, we need to find the value of sin(theta1) to solve for r.
To find the value of sin(theta1), we can use the inverse trigonometric function of sine, which is arcsin. We have:
sin(theta1) = 1.33 * sin(90 degrees)
theta1 = arcsin(1.33 * sin(90 degrees))
By evaluating this expression, we can find theta1.
Once we have theta1, we can find r using trigonometry.
Let's assume that the distance from the point on the surface directly above the fish to the boat is represented by r. Also, let's assume that the vertical distance from the surface to the fish is represented by h.
Using the definition of sine, we have:
sin(theta1) = h / r
Rearranging the equation, we get:
r = h / sin(theta1)
Now, you can substitute the value of theta1 that you found earlier, along with any known values for h, into the equation to solve for r.