Answer to September 4, 2000 Problem

adapted from a problem in
Twenty Years Before the Blackboard by Michael Steuben

Hawaiian Palm Tree Problem

 
There is a single 21-foot high vertical palm tree growing in the middle of a large flat volcanic Hawaiian 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 calculus teacher, who was visiting the island, lay down the night before this special day, and was awakened in the morning when the sun rays reached his eyes, which were 13 feet due west from the tree.

What time did he awake? Round your answer to the nearest minute.

 

Solution to Problem:

Answer is 9:53 AM.

Solution:
(1) Draw the figure by constructing a right triangle with vertical side equal to 21 and base equal to 13.
(2) Recognize that the time of day before noon is directly associated with the angle of elevation of the sun above the horizon (which equals the acute ange adjacent to the base of the triangle).
(3) Calculate the angle by taking the arctan (21/13) = 58.24 degrees.
(4) Using direct proportions, change the angle into hours and minutes:
58.24 / 180 = x hours / 12 hours => x = 3.88 hours.
.88 hours / 1 hour = y minutes / 60 minutes => y = 52.8 minutes.
So the answer is 6:00 AM + 3 hours + 53 minutes, or 9:53 AM.


Correctly solved by:

1. Joanna Castiglia Winchester, VA
2. David Powell Winchester, VA
3. Sarah Taylor Winchester, VA