Minimization

Date: 10/29/96 at 0:35:16
From: Andy Clark
Subject: Minimum Distance

My question deals with general applications of differentiation:

At noon, ship A is 100 miles due east of ship B.  Ship A is sailing 
west at 12 miles per hour and ship B is sailing south at 10 miles per 
hour.  At what time are the ships nearest to each other? How close are 
they?

I tried drawing a diagram, but my diagram is lacking information 
(or so it seems). Please help!

Andy Clark

Date: 10/29/96 at 6:6:58
From: Doctor Kate
Subject: Re: Minimum Distance

Andy,

I'll get you started. First, remember that what you're looking for is 
the time when distance is at a minimum. You should immediately look 
for a diagram that uses some 'x' in hours to express distances. This 
way you will be able to make an equation using time to express 
distance. So I let 'x' be the number of hours since noon.  Here's my 
diagram:

          <-------------------100------------------>
          *<-----100-12x------>A-------------------*
          ^                   /
          |                 /
          |               /
          |            /
         10x         /
          |        /
          |     /
          |   /
          ' /
          B

I'll explain the markings.  The *'s represent the original positions 
of the boats. I've marked on the top that they were originally 100 
miles apart. But 'x' hours have passed, and A has moved west and B has 
moved south. Now, B is 10x miles south of its original position 
(because it is moving at 10 miles per hour). Also, A is 12x miles west 
of its original position, or 100-12x miles east of B's original 
position (marked on the diagram) because it is moving at 12 miles per 
hour. Don't worry that 100 - 12x might get negative!  It will just 
mean that A is west of B's original position (remember -2 miles east = 
2 miles west).

Using the Pythagorean theorem:

   distance^2 = (10x)^2 + (100 - 12x)^2

Wait!  Don't square root - there's no reason to! You want a minimum 
distance, and since the distance will never be negative between A and 
B, distance will be a minimum when distance^2 is a minimum, and 
(10x)^2 + (100 - 12x)^2 is a lot easier to differentiate than the 
square root of all that stuff.

You should be able to do the minimization stuff from here without my 
help, but feel free to ask if anything causes trouble for you.

Hope this has helped,

-Doctor Kate,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math