$(4)$ Joan quit her job at GM when she was $35$ years old. Her friend, who is a financial advisor, recommended that she leave her $401($k$)$ retirement savings in GM$\text{'}$s retirement plan rather than withdrawing or transferring the money to a new plan. Her friend said that, on average, Joan could expect a $6\%$ increase per year if she left her money in the plan for many years, based on the past performance of GM$\text{'}$s plan. Joan will not be able to add more money to the GM $401($k$)$ account, but can open a new one at her new job.

$($a$)$ Create a model for the amount of money Joan will have in her $401(k)$ after any number of years. Use A for the accumulated amount of money, $t$ for the number of years, and $P$ for the amount of money in the plan when she quit, which is called the principal.

$($b$)$ What percent of the principal will Joan have if she leaves the money in the account for the $32$ years until she retires? What percent increase does this represent? Round each answer to the nearest hundredth of a percent.

$($c$)$ If Joan's original principal was $$20,100,$ how much would be in the account when she retires? Round to the nearest cent.

$($d$)$ Approximately how many years will it take the principal to double? How long to quadruple? Write your answers rounded to the nearest year.