Chapter 3.2, Problem 12E

For the logistic curve( $15$), assume $p_a:=p\left(t_a\right)$ and $p_b:=p\left(t_b\right)$ are given with $t_b=2t_a\left(t_a>0\right)$. Show that

$p_1=\left[\frac{p_a{p}_b-2p_0{p}_b+p_0{p}_a}{p_a{}^2-p_0{p}_b}\right]p_a,$

$A=\frac{1}{p_1{t}_a}\ln \left[\frac{p_b\left(p_a-p_0\right)}{p_0\left(p_b-p_a\right)}\right]$.

[Hint: Equate the expressions ( $21$ ) for $p_0$ at times $t_a$ and $t_b$. Set $\chi =\exp \left(-A{p}_1{t}_a\right)$ and ${\chi }^2=\exp \left(-A{p}_1{t}_b\right)$ and solve for $\chi$. Insert into one of the earlier expressions and solve for $p_1$.]