What are P and P' representing?
Let x= x(t) be the concentration of alcohol in a person, so 0 less than or equal to x less than equal to 1. t be the time in hours and A, k be constants. The model has x(t) satisfying the ODE P' = kP, x'= (-kx/(A + x)) after a person has stopped absorbing alcohol. Assume that initially the person's alcohol concentration is 0.024, three times the legal limit in many states.
A) How long does it take for the person's concentration to fall within the legal limit of 0.008, assuming A=0.005 and K-0.01
B) Choose new values for the constant A,k and discuss how the conclusion changes. Hypothesize some personal characteristics, gender, body mass index (BMI) and age that could affect the conclusions and this could be modeled by the values of A,k.
C) What terms would you add to ODE P' = kP to model the situation where a person is still absorbing alcohol at a constant rate?
Well it's obviously not a previous representation of the second DE, because if so it would have been written as x' = kx.
But anyway, you need to solve the DE (it's separable), and then set to solve for t.