How Does Refraction Help Calculate the Depth of a Pool?

[Huskies213]

Homework Statement

The bottom edge of the opposite side of an in-ground pool (n=1.33) is just visible at an angle of 18 degrees above the horizon. If the pool is 2m wide what is the depth?


I think I'm stuck on trying to relate the refraction formulas as to determining depth. .. any tips or help?

 

[dlgoff]

Make a cross-section drawing (of the pool and air above it) with a ray going from the bottom edge of the pool on one side, to the top edge of the pool on the other side. From there, draw the refracted ray. Can you now see how to find the depth from the angle and distance given?

 

[m2003]

I would approach this problem by first understanding the concept of refraction and its relationship to determining depth. Refraction is the bending of light as it passes through different mediums, such as air and water. This bending of light can make objects appear to be in a different position than they actually are.

In this scenario, the bottom edge of the pool appears to be at an angle of 18 degrees above the horizon due to the refraction of light passing through the water. To determine the actual depth of the pool, we can use the law of refraction, also known as Snell's law, which states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the speed of light in the first medium to the speed of light in the second medium.

In this case, the first medium is air with a refractive index of 1, and the second medium is water with a refractive index of 1.33. We can rearrange the equation to solve for the depth of the pool, which is represented by the distance d in the diagram below:

sin(18) = (1/1.33)sin(r)

Where r is the angle of refraction, which we can find by subtracting the angle of incidence (18 degrees) from 90 degrees (since the light is entering the water at a right angle).

r = 90 - 18 = 72 degrees

Substituting this into the equation, we get:

sin(18) = (1/1.33)sin(72)

Solving for d, we get:

d = 2m x tan(72) = 6.72m

Therefore, the depth of the pool is approximately 6.72 meters. By understanding the concept of refraction and applying the law of refraction, we were able to accurately determine the depth of the pool.