| 1) ||
Now, to find the slope of the line and the circle at that point, I will use implicit differentiation:
| 2) ||
| 3) ||
| 4) ||
I also know that the slope of the line at that point must be:
| 5) ||
Setting 5) equal to 4) (since the slope of the line equals the slope of the circle at the point in question), we get
| 6) ||
And cross multiplying we get
| 7) ||
| 8) ||
Substituting equation 1) into the left side, we get:
| 9) ||
| 10) ||
| 11) ||
Now to solve for our x-coordinate, which is a bit messy:
| 12) ||
| 13) ||
| 14) ||
Note to self:ï¿½suppress R2D2 jokeï¿½ï¿½.Dï¿½oh!
| 15) ||
| 16) ||
Okay, now we have the coordinates of that point, and we can find the angle.ï¿½ Using the following
sketch, we see that
| 17) ||
Finally for the fun part, letï¿½s plug some numbers in. First note that if d=0, the extra angle is
also zero, which is exactly what weï¿½d expect.ï¿½Also, we see that
| 18) ||
Just as weï¿½d expect. Using the radius of the Earth:
and mountain height:
which is the height of the Mauna Kea, one of five volcanic masses making up the 'Big Island' of Hawaii, (note that 13,796 ft or 4,205 m is the its part above the sea level - otherwise, from its base underneath the ocean to the top it is about 30,000 ft or 9,000 m tall, making it the tallest in the world when measured as such). I calculate
which translates into 8.32 minutes of extra time for viewing the sunset.
you can see that a 6 foot tall person standing up has an extra
over a man whoï¿½s eyes are at sea level, which translates into approximately 10 seconds of sunset.ï¿½ Of course, this small an angle could easily be obstructed by small things on the surface of the Earth, e.g. waves in the ocean.
Gregory Ogin, Physics Undergraduate Student, UST, St. Paul, MN
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