SOLUTION: An island is at point A, 4 km offshore from the nearest point B on a straight beach. A woman on the island wishes to get to go to a point C, 6 km down the beach from B. She can go

Algebra ->  Rational-functions -> SOLUTION: An island is at point A, 4 km offshore from the nearest point B on a straight beach. A woman on the island wishes to get to go to a point C, 6 km down the beach from B. She can go      Log On


   

Question 1176606: An island is at point A, 4 km offshore from the nearest point B on a straight beach. A woman on the island wishes
to get to go to a point C, 6 km down the beach from B. She can go by rowboat at 5 km/hr to a point P between B
and C and then walk at 8 km/hr along a straight path from P to C. Where should point P be located so the woman
can go to point C with the least possible time?
Answer by ikleyn(52788) About Me  (Show Source):
You can put this solution on YOUR website!
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An island is at point A, 4 km offshore from the nearest point B on a straight beach. A woman on the island wishes
to get to go to a point C, 6 km down the beach from B. She can go by rowboat at 5 km/hr to a point P between B
and C and then walk at 8 km/hr along a straight path from P to C. Where should point P be located so the woman
can go to point C with the least possible time?
~~~~~~~~~~~~~~ 


Let x be the distance from the point B to the point C.


Then the woman should go  sqrt%284%5E2+%2B+x%5E2%29 kilometers by rowboat at the speed of 5 kn/h

                  and walk (6-x) kilometers at the speed of 8 km/h.


The total time is  

          t(x) = sqrt%284%5E2%2Bx%5E2%29%2F5 + %286-x%29%2F8 hours.



To find the minimum of t(x), we should take the derivative of t(x) and equate it to zero

          0 = t'(x) = %282x%29%2F%282%2A5%2Asqrt%284%5E2+%2B+x%5E2%29%29 - 1%2F8 = x%2F%285%2Asqrt%2816%2Bx%5E2%29%29 - 1%2F8.


We then get this equation

          x%2F%285%2Asqrt%2816%2Bx%5E2%29%29 = 1%2F8

          8x = 5%2Asqrt%2816%2Bx%5E2%29

          64x^2 = 25*(16+x^2)

          64x^2 = 400 + 25x^2

          39x^2 = 400

          x = sqrt%28400%2F39%29 = 3.203 kilometers (approximately).


ANSWER.  The point P is located sqrt%28400%2F39%29 = 3.203 kilometers (approximately) down the beach from B.

Solved.