On $3/9/2013$ a $30-$year corporate bond was issued with a coupon of $3\mathrm{.}5\%.$

a$.$ What would be the bond's quoted price on $\frac{9}{9}\frac{?}{2016}$ if the YTM is $3\mathrm{.}5\%?$ If the YTM is

$3\mathrm{.}0\%?$ If the YTM is $4\mathrm{.}0\%?100\mathrm{.}000$; $109\mathrm{.}096$; $91\mathrm{.}876$

b$.$ What would be the bond's quoted price on $\frac{9}{9}\frac{?}{2022}$ if the YTM is $3\mathrm{.}8\%?$ If the YTM is

$4\mathrm{.}2\%?95\mathrm{.}974$; $90\mathrm{.}442$

c$.$ What would be the bond's price on $\frac{3}{9}\frac{?}{2024}$ if the YTM is $3\mathrm{.}2\%?$ If the YTM is $4\mathrm{.}1\%?$

$$1,042\mathrm{.}46$;$$921\mathrm{.}34$

Please explain the work for each question, especially how to figure out the t value used for time in the formula. I know it is $20$ but why?