Biologists studying a certain population of triglobytes have observed that the popula- tion fluxuates in a cyclical pattern. The minimum population size appears to be 1200 individuals. The last time the minimum was observed was January 1st, 2014. The maximum population size appears to be 2600 individuals. The last time the maximum was observed was September 1st, 2015. The biologists assume that the population can be modeled using a sinusoidal function T(x), where T is the population size measured in hundreds of individuals and x is time measured in months since January 1st, 2014.

(a) Determine the next time the population will be at its lowest level. Determine the next time the population will be at its highest level.

(b) Find a formula for T(x).

(c) Between January 1st 2016 and January 1st 2020, how many months is the population above 1900 individuals?

(d) Algebroans are a natural predator of triglobytes. The population of algebroans fluxuates in a cyclical pattern that is correlated with the triglobyte pattern. The minimum population size for algebroans is 250 individuals and the maximum population size is 900 individuals.

The lows and highs for the algebroan population occur 3 months after the corresponding lows and highs for the triglobyte population. Find a formula for A(x), the algebroan population x months after January 1st, 2014.

(e) In the same graph, sketch the first two periods (after January 1st, 2014) of the algebroan and triglobyte populations. Make sure you label the graph clearly. Include the midline for each population and the coordinates for the relative extrema for each function.