$($Optional bonus problem$)$ Air has a small, usually negligible index of refraction. It is $1\mathrm{.}0002926.$ This causes the Sun to actually be below the horizon when it appears to be just on the verge of sinking below it$.$ Suppose you are on the sea$-$shore watching the Sun apparently sinking into the ocean. When only its upper tip is still visible, by what fraction of the diameter of the Sun is that tip actually already below the surface?

To answer this question, as an approximation, take the earth's atmosphere as being of uniform density out to a thickness of $8\mathrm{.}70$ km $,$ beyond which there is no atmosphere. This means that, with the Earth's radius being $6371$ km $,$ your line of sight due West along the ocean surface to the horizon will intersect this "upper surface" of the atmosphere at about $333\mathrm{.}1$ km from your eye. $($If you enjoy trigonometry you can check this figure for yourself.$).$ Note that the diameter of the sun subtends $0\mathrm{.}5000$ degrees at your eye.