Question 1176606
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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?
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<pre>
Let x be the distance from the point B to the point C.


Then the woman should go  {{{sqrt(4^2 + x^2)}}} 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(4^2+x^2)/5}}} + {{{(6-x)/8}}} hours.



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

          0 = t'(x) = {{{(2x)/(2*5*sqrt(4^2 + x^2))}}} - {{{1/8}}} = {{{x/(5*sqrt(16+x^2))}}} - {{{1/8}}}.


We then get this equation

          {{{x/(5*sqrt(16+x^2))}}} = {{{1/8}}}

          8x = {{{5*sqrt(16+x^2)}}}

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

          64x^2 = 400 + 25x^2

          39x^2 = 400

          x = {{{sqrt(400/39)}}} = 3.203 kilometers (approximately).


<U>ANSWER</U>.  The point P is located {{{sqrt(400/39)}}} = 3.203 kilometers (approximately) down the beach from B.
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Solved.