Calculus Word Problem?

Question

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
$c (t)$ $c(t)$ . Assume that the function c obeys the following differential equation:

$\frac{d c}{d t}$ $(dc)/dt$ = $K_max$ $c/(k+c)$ $−rc$

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.

What is the maximal rate at which the substance is produced?

At what concentration is the production rate 50% of this maximum value?

If the production is turned off, the substance decays. How long will it
take for the concentration to drop by 50%?

At what concentration does production balance consumption?

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Answer 1 · 2016-12-16T18:21:39

See below.

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.

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