I'm stuck on these two problems....any help would be greatly appreciated!
1.) The amount of time a migrating bird can fly depends on its velocity, that is it is some function T(v). If E is the birdís initial energy, then the birdís effective
power is given by kE / T , where k is the fraction of the power that can be converted into mechanical energy. According to principles of aerodynamics, kE = aSv^3 + I , where a is
a positive constant, S is the wind speed, and I is the induced power, or rate of working against gravity. Find the velocity that a bird would need to fly to migrate a maximum distance (this will depend on some, but not all, of the parameters listed above).
2.) Standard population models allow us to predict the population of a species one year depending on the previous yearís population. One such model determines the population of salmon next year by the function f (S) = Se^r(1−S / P), where S is this yearís salmon population, P is the natural equilibrium population, and r is a constant that depends upon how fast the population grows. For different population sizes S, different amounts of salmon can be fished (harvested) in a year so that the population remains the same size. Such a harvest is sustainable over time. Find the maximum sustainable harvest. Do this by first finding the population S0 which will support the maximum sustainable harvest (this will depend on r and P). Then find the size of such a harvest (which will depend on S0 , r, and P).