Minimizing the Time to Rescue

Date: 11/29/1999 at 20:21:30
From: Chris Palmquist
Subject: Calculus (A related rate and optimization problem)

I'm standing on the beach at a certain point - let's call this point 
(0,0). There is a man drowning in the water 300 meters down the beach 
and 100 meters out. Call his point (300, 100). I can run 5 meters per 
second and can swim 3 meters per second. How far should I run along 
the beach before jumping in the water and swimming to the drowning 
man in the fastest time?

I've tried everything on this problem and I just can't get it. I know 
you need to use the related rate to find the minimum, but I can't 
really get started on this question. Please help me out.

Date: 11/30/1999 at 08:13:54
From: Doctor Jerry
Subject: Re: Calculus (A related rate and optimization problem)

Hi Chris,

Let the beach be the x-axis with your present position the origin. You 
want a point x, 0 <= x <= 300, for which the sum of TW+TS is a 
minimum, where TW is the time walking from (0,0) to (x,0) and TS is 
the time spend swimming from (x,0) to (300,100). Each is a constant 
rate situation and so D = R*T, or T = D/R. 

I hope that will get you started.

I think it's a mistake to try to over-categorize a problem before 
dealing with it directly. I'm referring to your feeling that it is a 
related rates problem.

- Doctor Jerry, The Math Forum
  http://mathforum.org/dr.math/