OK---i have a problem.

you lie on a beach and watch a sunset. then you stand and watch it set again--you are higher and can see that supposedly.

my book says that you then use these values:

$\displaystyle h=1.70 m$ (your height)

$\displaystyle t=11.1 s$ (the time between sunsets)

then using the picture and the Pythagorean theorem, with $\displaystyle d$ as the distance between your eyes and the tangent to the earth from the second sunset, you get

$\displaystyle d^2 + r^2 = (r + h)^2 = r^2 + 2rh + h^2$

or

$\displaystyle d^2 = 2rh + h^2$

but you remove $\displaystyle h^2$ because your height is insignificant compared to the radius of the earth and the term $\displaystyle 2rh$

then you use the number of hours in a day, the rotation of the earth in a day, and $\displaystyle t$ to find $\displaystyle \theta$:

$\displaystyle \frac \theta {360^o}$

which solves to $\displaystyle 0.04625^o$

then you can get $\displaystyle d = r\tan \theta$ from the picture and, substituting for $\displaystyle d$ in the first equation, $\displaystyle r^2\tan^2\theta = 2rh$

which when you plug in the values (like $\displaystyle \theta = 0.04625$) makes $\displaystyle 5.22 * 10^6 m$

The problem is in the fact that we replaced the value for the $\displaystyle \theta$ from the angle between the sunsets with the $\displaystyle \theta$ from the angle from the center of the earth. i cant figure out how that works.

Here is the picture: