Suppose that a country's population is $42$ million and is currently growing at a rate of $11\%$ per year. If the population growth follows a logistic growth model with a carrying capacity of $82$ million, calculate the annual growth rate when the population is $48$ million.

$11\mathrm{.}3\%\%$

$7\mathrm{.}6\%\%$

$9\mathrm{.}8\%\%$

$9\mathrm{.}4\%\%$