Richard writes:
> Now this involves a little bit of calculus - If Painter A and Painter B
> both have different colors, (Assume Immediate drying time), How long
> would it take Painter A to have the room completely painted?  Assuming
> no fowl play by either painter.

Oh, all right.  Let me give it a shot.  I'm probably missing something, but
this looks like a pretty simple problem.

For simplicity and tradition we'll assume a cylindrical room (and a
spherical cow should one be required).

We assume the painters are working in the same direction around the room
(i.e. clockwise when viewed from above) otherwise they'll run out of paint
long before the room is all one color.

We have:

   RateA = 1/3 room/hour; RateB = 1/4 room/hour;

The amount of room painted by each will be equal to the painting time 't'
multiplied by their rate.

We have to decide the relative "start" positions of the two painters in the
room.

If, say, they start at the same point on the same wall, (not very efficient)
then we want to know the time t when the number of rooms painted by A is
exactly one greater than the number of rooms painted by B.  So we want to
know when

   t * RateA = t * RateB + 1

Being really amazingly lazy we ask Mathematica:

   RateA = 1/3 room/hour; RateB = 1/4 room/hour;

   Solve[t RateA == t RateB + 1 room, t]

   {{t->12 hour}}

So, after 12 hours, A has caught up to B (and there will be exactly seven
coats of paint on the wall at that point).  This is also the worst-case time
for Painter A to catch up to Painter B.

If they start out on opposite sides of the room (180 degrees apart on the
perimeter of our cylindrical room) then Painter A only needs to advance half
of a room farther than the other painter (as long at least one complete coat
of paint is applied by Painter A during that interval so the minimum time
will be three hours no matter how slow Painter B is).  And sure enough:

   Solve[t RateA == t RateB + 1/2 room, t]

   {{t->6 hour}}

So it only takes six hours for Painter A to catch Painter B.  Of course
Painter b has only applied 1 1/2 coats of paint to Painter A's 2 coats, so
the room has an uneven 3.5 coats of paint.

G.

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