Confused on the setup of the Differential problem and workout

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?

Re: Confused on the setup of the Differential problem and workout

What are P and P' representing?

Re: Confused on the setup of the Differential problem and workout

I think the P and P' is just the former representation of the x'= - kx/A + x equation. I'm a little confused on that as well.but I'm assuming that this is what they mean.

Re: Confused on the setup of the Differential problem and workout

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 $\displaystyle \displaystyle \begin{align*} \frac{dx}{dt} = \frac{-0.01\,x}{0.005 + x} \end{align*}$ (it's separable), and then set $\displaystyle \displaystyle \begin{align*} x = 0.08 \end{align*}$ to solve for t.