Question #7

Explain in words, using appropriate variable names, how you would solve the problem of fish population vs. clock time given the equation for the P`, the rate of change of population, in terms of the population.

Initial Population will equal 1stP.  First we want  to find the change in P to find the population of the fish at the end of the time interval, so we use the equation 1stP * P` = 1stChangeP.  To find what the population of the fish at the end of the clock time is we add 1stP + 1stChangeP which equals 2ndP (1stP + 1stChangeP = 2ndP).  However, if we use the formula 2ndP * P` = change in P we get a different change in P, or 2ndChangeP.  So to find the average rate at which 1stP increases for a more accurate number, first add 2ndChangeP and 1stChangeP, then divide by two.  This gives us the average change in P, or AveChangeP.  Set up the equation 1stP * P` = AveChangeP.  Using algebra, the equation can be changed to P` = AveChangeP/1stP.  This equation allows us to find the average rate of increase in the fish population over the given period of time.  Substitute this average rate into the original equation 1stP * AveP` = 2nd P.  This will give us a more accurate number for the fish population after the given period of time.