Calculus Optimization Swimmer Problem

Question

A swimmer is at a point 500 m from the closest point on a straight shoreline. She needs to reach a

cottage located 1800 m down shore from the closest point. If she swims at 4 m/s and she walks at

6 m/s, how far from the cottage should she come ashore so as to arrive at the cottage in the shortest

time?

Answer 1

The equation is derived from the total time = time swimming to shore at x + time walking from x to 1800 m along shore. In terms of distance and velocity t = d/v and the total time becomes

t = sqrt(500² + x²)/4 +(1800-x)/6

Take derivative of t with x and set to 0 to find extrema:

t = (1/2)(2x)/(2500+x²)^1/2 - 1/6 = (x/4)/(2500+x²)^1/2 - 1/6 = 0

x = 44.721 using graphing.for 0 (You can solve using quadratic formula by adding 1/6 to both sides, squaring, and putting into standard quadratic form)

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