$4-30.$ Last week Evelyn moved into the new house she purchased, and she is already making plans to pay off the mortgage early. To purchase the house, she borrowed $$280,000$ with terms that require her to make monthly payments for $15$ years and pay interest equal to $5\mathrm{.}4$ percent, compounded monthly.

$($a$)$ What is Evelyn's monthly mortgage payment?

$($b$)$ How much will Evelyn owe on the mortgage after she has made payments for five years; that is$,$ when there are $10$ years of monthly payments remaining?