Manuel wants to invest an inheritance of $$20,000$ for one year. His credit union offers $5\mathrm{.}13\%$ for a one$-$year term or

$5\mathrm{.}08\%$ for a six$-$month term.

$($a$)$ How much will Manuel have after one year if he invests at the one$-$year rate?

$($b$)$ How much will Manuel have after one year if he invested for six months at a time at $5\mathrm{.}08\%$ each time?

$($c$)$ What would the one$-$year rate have to be to yield the same amount of interest as the investment described in

part $(b)?$

$($a$)$ After one year, Manuel will have $

$($Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places

as needed.$)$

$($b$)$ One year after the initial investment, Manuel will have $

$($Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places

as needed.$)$

$($c$)$ The one$-$year rate to yield the same amount of interest as the investment described in part $($b$)$ is $\%.$

$($Round the final answer to two decimal places as needed. Round all intermediate values to six decimal places

as needed.$)$