A $30-$year maturity bond making annual coupon payments with a coupon rate of $11\mathrm{.}00\%$ has a Duration of $13\mathrm{.}50$ vears. The bond currently sells at a yield to maturity of $5\mathrm{.}75\%.$

a$.$ Find the exact dollar price of the bond if its yield to maturity falls to $4\mathrm{.}75\%.$ What is the $\%$ change in price?

b$.$ Assume that you need to make a quick approximation using the duration rule.

What is the $\%$ change in price as approximated by the duration rule when the yield to maturity falls to $4\mathrm{.}75\%?$

c$.$ Does the duration$-$rule provide a good approximation of the $\%$ price change in this case? Why or why not?