Tetanus bacillus bacterium is cultured to produce tetanus toxin used in an inactive form for the tetanus vaccine. the amount of toxin produced per batch increases with time and then becomes unstable. the amount of toxin (in grams) as a function of time t (in hours) can be approximated by: b(t) = - 1/1426 t^2 + 1/31 t how many hours will it take to produce the maximum yield? the maximum yield will occur in hours.

The biologist has a second culture to examine. she knows that the population of the culture doubles every 15 minutes. after 1 hour and 15 minutes, her assistant found that 80,000 bacteria were present. a) what was the size of the initial population? b) predict the size of the culture at t=3 hours. what was the size of the population at 40 minutes? explain and justify your answers. c) create a graph of the population as a function of time. find an equation that can be used to predict the size of the population at any time t. d) examine the rate at which the bacteria culture is growing. how fast is the culture growing after 1 hour? after 1.5 hours? after 2 hours? use a time interval of h = 0.01 hours to estimate these rates. interpret these rates in terms of the context of the problem situation. how do these three rates compare?