$3.$ Consider a $30-$year, $8\%$ bond that is trading at a yield of $10\%.$ The coupons are paid twice a year.

$($a$)$ Find the price

$($b$)$ Find the duration and the modified duration

$($c$)$ Suppose the yield has gone up by $1\%.$ Can you guess the current bond price using the information from $($b$)?$