Thursday, October 24, 2013

On Reduction, 4 of 11

Applications of Reduction

     The “Many Hands” Theorem:
     Assume that r1t1 = r2t2 = d. Then:
                d/(r1+r2)  =  (t1<+>t2)
                d/(t1+t2)  =  (r1<+>r2)
                d/(r1-r2)  =  (t1<->t2)
                d/(t1-t2)  =  (r1<->r2)
     In a rate-time problem, when rates add, times reduce; and when rates subtract, times protract.  “Many hands make light work.”

     If Alice can trim the hedges in A hours of work, and Bob can trim the hedges in B hours, then if they work together, and if their work rates add, then they will finish the job in A<+>B hours.

     Two ships cross the same ocean, starting at the same time and heading for each other's home port. Let F be the time that the first ship took to go from port A to port B, let S be the time that the second ship took to go from port B to port A, and let M be the time it took for them to meet in mid-ocean. Approach-rates add, so times reduce: M equals F <+> S.

     Two racecars, when driving towards each other from a distance of a mile, pass each other in P seconds; when the faster car pursues the slower car from one mile behind, it takes C seconds to catch up. How quickly does each car run a mile?
     Answer: Respectively,  2(P<+>C)  and  2(P<->C) seconds.

     If you drive a mile at velocity v1, and then another mile at velocity v2, then on average you will go at velocity 2(v1 <+> v2); the harmonic mean.

     Kirkhoff's Law for parallel resistors goes:
           R12P  =  R1 <+> R2 .
     Resistances reduce in parallel.

     Two simple lenses with focal lengths f1 and f2, when placed together, form a compound lens with focal length   f1 <+> f2 .

     A lens made of material with index of refraction n, and with lens radii r1 and r2, has focal length  (r1 <-> r2)/(n-1) .

                              .  . |                           
                          .    .   |
                      .      .     |                           
                  .        .       |  h
              .          .         |                           
          .            .           |
      .   a          . b           |
      A       x     B              C

     In this diagram,     h   =   x (tan(a) <-> tan(b))

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