a)
To find the inverse function we need to solve the expression for t so let's do that:
[tex]\begin{gathered} q=k(1-e^{-\frac{t}{a}}) \\ 1-e^{-\frac{t}{a}}=\frac{q}{k} \\ e^{-\frac{t}{a}}=1-\frac{q}{k} \\ \ln e^{-\frac{t}{a}}=\ln(1-\frac{q}{k}) \\ -\frac{t}{a}=\ln(1-\frac{q}{k}) \\ t=-a\ln(1-\frac{q}{k}) \end{gathered}[/tex]
Therefore, the inverse function is:
[tex]t=-a\ln(1-\frac{q}{k})[/tex]
This function will tell us the time it takes the capacitor to store a charge q.
b)
To charge the capacitor to a 90 percent capacity means that q=0.9k; this comes from the fact that maximum charge is k. Plugging this and the value of a we have:
[tex]\begin{gathered} t=-2\ln(1-\frac{0.9k}{k}) \\ t=-2\ln(1-0.9) \\ t=4.6 \end{gathered}[/tex]
Therefore, it takes the capacitor 4.6 seconds to charge to ninety percent of its capacity.
