Question

A small island is 4 miles from the nearest point P on the straight shoreline of a large lake. If a woman on the island can row a boat 2 miles per hour and can walk 3 miles per hour, where should the boat be landed in order to arrive at a town 7 miles down the shore from P in the least time? Let be the distance between point P and where the boat lands on the lakeshore. Hint: time is distance divided by speed.

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Answer 1

is
\[\frac{\sqrt{16+x^2}}{2}+\frac{7-x}{3}\] right?

Answer 2

if so, take the derivative, set it equal to zero and solve

Answer 3

derivative is
\[\frac{x}{2\sqrt{x^2+16}}-\frac{1}{3}\]

Answer 4

solve
\[\frac{x}{2\sqrt{x^2+16}}=\frac{1}{3}\] for \(x>0\)

Answer 5

i think you get \(x=\frac{8}{\sqrt{5}}\)