Answer
$a = 140$
$b = 6$
In the long run, the yeast population will be 140
Work Step by Step
$f(t) = \frac{a}{1+b~e^{-0.7t}}$
We can find an expression for $f'(t)$:
$f(t) = \frac{a}{1+b~e^{-0.7t}}$
$f'(t) = \frac{-(a)(b~e^{-0.7t})(-0.7)}{(1+b~e^{-0.7t})^2}$
$f'(t) = \frac{0.7ab~e^{-0.7t}}{(1+b~e^{-0.7t})^2}$
It is given that $f(0) = 20$ and $f'(0) = 12$
$f(0) = \frac{a}{1+b~e^{-0.7(0)}} = \frac{a}{1+b} = 20$
$a = 20(1+b)$
We can find the value of $b$:
$f'(0) = \frac{0.7ab~e^{-0.7(0)}}{(1+b~e^{-0.7(0)})^2} = \frac{0.7ab}{(1+b)^2} = 12$
$\frac{0.7b~[20(1+b)]}{(1+b)^2} = 12$
$\frac{14b}{1+b} = 12$
$14b = 12b+12$
$2b = 12$
$b = 6$
We can find the value of $a$:
$a = 20(1+b)$
$a = 20(1+6)$
$a = 140$
We can find the value of $f(t)$ in the long run:
$\lim\limits_{t \to \infty}f(t) = \lim\limits_{t \to \infty}\frac{a}{1+b~e^{-0.7t}}$
$\lim\limits_{t \to \infty}f(t) = \frac{a}{1+b~(0)}$
$\lim\limits_{t \to \infty}f(t) = a$
$\lim\limits_{t \to \infty}f(t) = 140$
In the long run, the yeast population will be 140
