there is a single 21-foot vertical palm tree growing in the middle of a large flat desert island.

Question

there is a single 21-foot vertical palm tree growing in the middle of a large flat desert island. On a certain day, the sun will rise at 6:00 AM and set at 6:00 PM. At noon on that day the sun will be directly overhead and the trunk of the palm tree will cast no shadow. A traveler lay down the night before this special day, and was awakened in the morning when the sun's rays reached his eyes, which were 13 feet due west from the tree. To the nearest minute, at what time did he awake?

Answer 1

The sun goes 180 degrees in 12 hours. So it goes 180/12=15 degrees per hour or 15 degree per 60 minutes, or 0.25 degree per minute. The traveler awoke when the sun was at an angle

A=arctan(21/13)≈58.24º above the horizon. It took the sun 58.24(degrees)/0.25(deg/min)≈233 minutes. Or 3 hours 53 minutes. Since sunrise was at 6 a.m., he awoke at 9:53 a.m.