Expert: Paul Klarreich - 10/18/2010

Question

George takes a trip from St. Louis to Chicago.  He leaves at 9am on Monday and arrives at 2pm that day.  He returns on Tuesday, leaving at 9am and arriving back in St. Louis at 2pm, retracing exactly the same route.  Show that there is a point on the road through which he passes at the same time both days.


Answer
Questioner:   Jeanne
Category:  Calculus
Private:  No
 
Subject:  calculus
Question:  
George takes a trip from St. Louis to Chicago.  He leaves at 9am on Monday and arrives at 2pm that day.  He returns on Tuesday, leaving at 9am and arriving back in St. Louis at 2pm, retracing exactly the same route.  Show that there is a point on the road through which he passes at the same time both days.
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I have taken the liberty of rewording your question slightly:

At 9:00 A.M. a medium-suede Grand Caravan with a mommy, a daddy, and their three daughters, leaves Brunswick, GA, drives South on I95 at varying speed, and reaches Miami, FL at 5:00.  

A week later, they leave Miami at 9:00 AM, drive North on I95 at varying speed, and reach Brunswick, GA at 5:00.

Is there a point between Miami and Brunswick which they pass at exactly the same time on both trips?

That is exactly how I posed the question to my daughters, Naomi and Erica on one of our yearly trips.
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Naomi [PhD, Rutgers] answered: Yes, because:

The position P1(t) of the car, expressed as distance from Miami, on the Southbound trip has the properties that:

P1(t) is continuous.
P1(9:00) = 411 miles.
P1(5:00) = 0

Likewise, P2(t), for the Northbound trip, has:

P2(t) is continuous.
P2(9:00) = 0
P2(5:00) = 411 miles.

And P(t) = P2 - P1, the distance between the cars, is:

P(t) is continuous.
P(9:00) =  - 411
P(5:00) = + 411 miles.

By the Intermediate Value Theorem, there is a time t, between 9 and 5, where P = 0.  That is the time.
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Erica [PhD, Stony Brook] answers:  The same.
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Daddy [no PhD, just an old math teacher] answers:  Yes, because:

At 9:00 A.M. a medium-suede Grand Caravan with a mommy, a daddy, and their three daughters, leaves Brunswick, GA, drives South on I95 at varying speed, and reaches Miami, FL at 5:00.  

AT THE SAME TIME, another medium-suede Grand Caravan with a mommy, a daddy, and their three daughters, leaves Miami at 9:00 AM, drives North on I95 at varying speed, and reaches Brunswick, GA at 5:00.

There must be a time when those two cars pass each other, right?
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Are you sure you don't know some members of my family?  Did you really get this yourself?