Question

Calculus Word Problem?

Consider a biochemical reaction in which a certain substance is both produced
and consumed. The concentration of this substance at time t is defined to be
`c(t)`. Assume that the function c obeys the following differential equation:

`(dc)/dt` = `K_max` `c/(k+c)` `−rc`

Where `K_max`, `k`, and `r` are all positive constants. The first term on the right
hand side of this equation denotes the concentration-dependent production
and the second denotes the consumption.

1. What is the maximal rate at which the substance is produced?
2. At what concentration is the production rate 50% of this maximum value?
3. If the production is turned off, the substance decays. How long will it
   take for the concentration to drop by 50%?
4. At what concentration does production balance consumption?

Answer 1

See below.

Explanation:

1) The rate of produced substance is maximum when

`K_(max)c/(k+c)` is maximum. This function is monotonic increasing so it's maximum is

`lim_(c->oo)K_(max)c/(k+c)=K_max`

2) Choosing

`0.5K_(max)=K_(max)c/(k+c)`

and solving for `c` we obtain

`c = k`

3) If production is turned off the the concentration evolution is given by

`(dc)/(dt)=-r c` with solution

`c = C_0 e^(-r t)` so the concentration will drop by `50%` when

`1/2=e^(-rt)` or after solving for `t`

`t = log_e(2)/r` seconds.

4) The balance is attained when

`K_(max)c/(k+c)-rc=0` or when production equals consumption.

`c = (K_(max) - k r)/r`