(a)[3]
Show that $p$ fulfils the equation $\ln(5p)=\frac{1}{5p}$.
(b)[2]
By drawing a suitable pair of graphs, show that the equation in part (a) has just one root.
(c)[2]
Show by calculation that $0.2 \le p \le 0.6$.
(d)[3]
It is given that the equation in part (a) may be rewritten as $p=\frac{1}{5}\exp\!\left(\frac{1}{5p}\right)$, where $\exp(x)$ means $e^x$. Use an iterative formula based on this rearrangement to calculate $p$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.