Answer:
a) 404 m² b) apparent height = 7.5 m
Explanation:
This question is about refraction and total internal refraction.
Here I will take refractive index of air and water
![n_{air}=1\\ n_{water}=1.33=4/3](https://tex.z-dn.net/?f=n_%7Bair%7D%3D1%5C%5C%20n_%7Bwater%7D%3D1.33%3D4%2F3)
Now let's look at the diagram I have attached here
At some angle A, the light from the ring (yellow point) under water will be totally internally refracted (B = 90°), which means that rays of light (yellow arrow) that make large enough angle A will not be able to escape from the water. Since we assumed that the ring is a point, there will be a critical cone of angle A with the ring at its apex which traces a circle of radius R on the surface of water, which, beyond this radius, no light could escape.
According to snell's law
![\frac{sin(B)}{sin(A)} = \frac{n_{water}}{n_{air}} = 4/3](https://tex.z-dn.net/?f=%5Cfrac%7Bsin%28B%29%7D%7Bsin%28A%29%7D%20%3D%20%5Cfrac%7Bn_%7Bwater%7D%7D%7Bn_%7Bair%7D%7D%20%3D%204%2F3)
At critical angle B = 90°
![\frac{3)}{4}sin(B) = [tex]\frac{3}{4} sin(90^\circ ) = 0.75 = sin(A)](https://tex.z-dn.net/?f=%5Cfrac%7B3%29%7D%7B4%7Dsin%28B%29%20%3D%20%5Btex%5D%5Cfrac%7B3%7D%7B4%7D%20sin%2890%5E%5Ccirc%20%29%20%3D%200.75%20%3D%20sin%28A%29)
Therefore
![A = 48.6^\circ](https://tex.z-dn.net/?f=A%20%3D%2048.6%5E%5Ccirc)
With this, we can find the radius of the circle (refer to my diagram)
![h* tan (A) = R\\R =11.3 m](https://tex.z-dn.net/?f=h%2A%20tan%20%28A%29%20%3D%20R%5C%5CR%20%3D11.3%20m%20)
And with that we can find the area
![A = \pi R^2=404\ m^2](https://tex.z-dn.net/?f=A%20%3D%20%5Cpi%20R%5E2%3D404%5C%20m%5E2)
Additional Problem
For apparent depth from above, we can think that, since we are accustomed to seeing light at the speed of c in air, our brain interpret light from <em>any</em> source to be traveling at c. This causes light that originated under water, which has the speed of
![v_{water} = \frac{c}{n_{water}} = 0.75c](https://tex.z-dn.net/?f=v_%7Bwater%7D%20%3D%20%5Cfrac%7Bc%7D%7Bn_%7Bwater%7D%7D%20%3D%200.75c)
to appear as if it has traveled with the same duration as light with speed c
In order for this to happen our brain perceive shortened length which is the apparent depth.
To put it in mathematical term
![t_{travel}=\frac{h_{apparent}}{v_{water}} =\frac{h}{c}](https://tex.z-dn.net/?f=t_%7Btravel%7D%3D%5Cfrac%7Bh_%7Bapparent%7D%7D%7Bv_%7Bwater%7D%7D%20%3D%5Cfrac%7Bh%7D%7Bc%7D)
So we get apparent depth
![h_{apparent}=0.75h = 7.5\ m](https://tex.z-dn.net/?f=h_%7Bapparent%7D%3D0.75h%20%3D%207.5%5C%20m)