Homework SourseA small island is 3 miles from the nearest point P on the st
A small island is 3 miles from the nearest point P on the straight shoreline of a large lake. If a woman on the island can row a boat 3 miles per hours and can walk 4 miles per hour, where should the boat be landed in order to arrive at a town 10 miles down the shore from p in the least time?
The boat should be landed ? miles down the shore from P
The boat should be landed ? miles down the shore from P
Solution
If the boat land x miles below P then, using the Pythagorean Theorem, the distance traveled by boat is sqrt(x^2+3^2) and obviously, the distance traveled by foot is 10-x. Therefore the time it takes is f(x) = sqrt(9+x^2)/3+(10-x)/4 The time is minimal when the derivative of f(x) is 0 f\'(x)=x/(3sqrt(9+x^2))-1/4=0 4x=3sqrt(9+x^2) 16x^2=9(9+x^2) 7x^2=81 x=9sqrt(7)/7 miles ~ 3.4 miles
