$2.$ Gordon Freeman wants to have $$32,000$ in $5$ years in order to purchase a new car.

$($a$)$ How much should he deposit today in an account earning $6\mathrm{.}4\%,$ compounded quarterly, to have the required amount in $5$ years?

$($b$)$ How much interest will be earned?

$($c$)$ If he can only deposit $$16,000$ now, how short of $$32,000$ will he be$?$

$($d$)$ Suppose he instead decides to make a deposit at the end of each quarter into an account earning, $6\mathrm{.}4\%,$ compounded quarterly. How much should he deposit each quarter to have $$32,000$ in $5$ years?