



Example:






Person 1
is standing at some point. 


Person 2
is standing 20 miles to the east of Person 1 ... 











At the
same moment, Person 1 starts walking south at 4 miles per hour, 


and
Person 2 starts walking west at 2 miles per hour ... 











How long
after they start walking, will the distance between them be a
minimum? 





OK, the
first thing to do is to figure out the equation 


for the distance
between them. 


We drop a
coordinate axis on them (I hope it didn't hurt much). 


It will
make things easier if we put the origin at the place person 1
started ... 











So by
arranging the axis this way, each person walks along one of the
axis. 


That is,
for person 1 position, only the Y coordinate changes 


for
person 2, only the X coordinate changes. 


You will
see in a second that that makes the math easier. 





Person 1
is moving south along the Y axis. 


When person 1 starts walking, they
are at (0,0). 


At any
time (T) after that, they are at: 





(0,
(04T))






Person 2
starts out at a point 20 miles east of person 1. That is, at (20,0) 


Person 2
walks west at 2 miles per hour. 


That
means at any point in time they are at: 





((20
 2T), 0)






If you
look at the diagram above, 


you can
see that the distance between them is the hypotenuse of a triangle. 


That
means we can use the old Pythagorean distance formula! 


So at any
time T, the distance between them is: 











Multiplying
this out ... 











Combining
terms ... 











We want
to find where the distance, D, is a minimum. 


That
means we need to check the point where the slope is zero and the
endpoints. 


One
endpoint is where time = 0, that is, the starting point. 


There
really isn't another end point. Time can run forever here. 


That
means we need to calculate a limit as time approaches infinity. 


To get
the slope equals zero point, 


we need the first derivative of the
distance equation. 





I find it
easier to change roots to fractional exponents when doing this. 


A square
root is the same as raising the expression to the one half power. 


So ... 











Don't
forget that this is a power chain rule problem, 


so we have to deal
with an "inside part." 





We need
to find the point where D'=0, 


so it would help to write this as a
real fraction... 











When the
numerator equals zero, the whole expression will equal zero. 


As long
as the denominator does not equal zero. 


So ... 


0
= 40T  80



T
= 2 hours






Check to
make sure the denominator is not equal to zero at T = 2. 


We only
need to find the value under the square root radical. 





20T^{2}
 80T + 400



20(2)^{2}
 80(2) + 400



80
 160 + 400



320






OK, the
denominator is not equal to zero. 


Now lets
check the second derivative to make sure that this is a minimum... 


Use the
quotient rule, and hold on ... 











Do some
simplifying ... 











Solve
this mess for when T = 2 ... 











The
second derivative is positive. 


That
means the T=2 hours point is a minimum. 





So what
is the distance between the two people at that point? 


The
distance formula was ... 











Plug in T
= 2 and chug away. 


You
should get ... 


D
= 17.89 miles






The
endpoints are where T = 0 and where T approaches infinity. 


At T = 0,
the first two terms under the radical equal zero 


so the
distance is the square root of 400, so 





D
= 20






Where T
approaches infinity, 


the value will be controlled by the highest
exponent term under the radical. 


It takes
a bit of fancy math to prove this one, 


but it is
pretty obvious that the distance approaches infinity. 





So the
minimum distance of 17.89 miles happens when T = 2. 





copyright 2005 Bruce Kirkpatrick 
