Expert: Paul Klarreich - 3/22/2009

Question
Henry, who is in a rowboat A 2 miles from the nearest point B on a straight shoreline, notices smoke billowing from his house C, which is 6 miles down the shoreline from B. He figures he can row at 6 miles per hour and run at 10 miles per hour. How should he proceed in order to get to his house in the least time?

Answer
Questioner:   Neka
Country:  Bahamas
Category:  Calculus
Private:  No
 
Subject:  Min/Max problems
Question:  Henry, who is in a rowboat A 2 miles from the nearest point B on a straight shoreline, notices smoke billowing from his house C, which is 6 miles down the shoreline from B. He figures he can row at 6 miles per hour and run at 10 miles per hour. How should he proceed in order to get to his house in the least time?
 
B    A       C
+------------+
|   /
|  /
| /
|/
R
Let AB = x, then AC = 6 - x

Then:  RB = 2,  

AR = sqrt(4 + x^2)

Total time(to be min'ed)
     RA      AC
T = ------ + -----
      6      10


   sqrt(4 + x^2)   (6-x)
T = ------------- + -----
      6             10

OK, you can do it from here. BUT be sure to check the endpoints.  Maybe direct rowing from R to C, or Rowing straight to B, is the answer.

AN AGAIN, BE SURE to check the archives.  Look for problems with

Maximum-minimum problem

as the subject line.  THERE ARE A TON OF THEM.