The academics and the technicians of a university faculty are planning a paintball battle and the mathematicians are trying to predict their chances of winning with a continuous model. They are representing the numbers in the two teams at any time t as and respectively and have decided that the system to model the encounter should be dA/dt= -k2 T and dT/dt= -k1 A, where and are positive constants.
State clearly what assumptions are implicit in formulation of the system and say what the constants k1 and k2 represent.
The muscles and eyesight of the academics have of course suffered from too much ‘book learning’ over the years, but at least they are aware of their limitations as good soldiers. They estimate that although both sides can fire paintballs at the same rate as each other, denoted by f_A= f_T=2 shots per minute, the technicians have a probability of 0.035 of hitting their target with a single paintball shot whilst the academics have only a 0.01 probability of hitting theirs. Based on the initial conditions , The system of ODEs can of course be solved analytically using eigenvalue techniques over a 20 minutes battle then comment on the outcome after that time.

Question

The academics and the technicians of a university faculty are planning a paintball battle and the mathematicians are trying to predict their chances of winning with a continuous model. They are representing the numbers in the two teams at any time t as   and  respectively and have decided that the system to model the encounter should be dA/dt= -k2 T   and dT/dt= -k1 A,  where  and are positive constants.
State clearly what assumptions are implicit in formulation of the system  and say what the constants  k1 and k2 represent.
The muscles and eyesight of the academics have of course suffered from too much ‘book learning’ over the years, but at least they are aware of their limitations as good soldiers. They estimate that although both sides can fire paintballs at the same rate as each other, denoted by f_A= f_T=2 shots per minute, the technicians have a probability of 0.035 of hitting their target with a single paintball shot whilst the academics have only a 0.01 probability of hitting theirs.  Based on the initial conditions , The system of ODEs can of course be solved analytically using eigenvalue techniques over a 20 minutes battle then comment on the outcome after that time.

anonymous · Answer

k1 and k2 represent how long is takes for each team to decrease by a factor of $e=2.718\dots $

anonymous · Answer

The assumption is that as each team grows smaller, it will take longer for the remainder of the team members to die, because they'll be harder to find.

anonymous · Answer

how did you get e=2.718?

anonymous · Answer

$e$ is just a constant.  It's the base of the natural log function.

anonymous · Answer

ok .Thanx