A rumor is spread in a school. For $0<a<1$ and $b>0$, the time $t$ at which a fraction $p$ of the school population has heard the rumor is given by $$t(p) = \int_{a}^{p} \frac{b}{x(1-x)}\,dx.$$

(a) Evaluate the integral to find an explicit formula for $t(p)$. Write your answer so it has only one $\ln$ term.
$\int_{a}^{p} \frac{b}{x(1-x)}\,dx =$

(b) At time $t=0$, two percent of the school population ($p=0.02$) has heard the rumor. What is $a$?
$a =$

(c) At time $t=1$, fifty-five percent of the school population ($p=0.55$) has heard the rumor. What is $b$?
$b =$

(d) At what time has ninety-one percent of the population ($p = 0.91$) heard the rumor?
$t =$