.
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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Let x be the distance from the point B to the point C.
Then the woman should go 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) = + hours.
To find the minimum of t(x), we should take the derivative of t(x) and equate it to zero
0 = t'(x) = - = - .
We then get this equation
=
8x =
64x^2 = 25*(16+x^2)
64x^2 = 400 + 25x^2
39x^2 = 400
x = = 3.203 kilometers (approximately).
ANSWER. The point P is located = 3.203 kilometers (approximately) down the beach from B.
Solved.