The couple in Problem $12$A$-37$ has only enough money for a $10\%$ down payment and other closing costs. Thus the bank is offering a loan at a higher rate of $5\mathrm{.}15\%$ and requiring private mortgage insurance, which costs $1\%$ of the loan amount each year. Contributed by Kate Abel, Stevens Institute of Technology

$($a$)$If the maximum debt$-$to$-$income ratio $($total monthly debt divided by after$-$tax monthly income$)$ is still $43\%,$ can the couple afford to purchase the home?

$($b$)$If the couple lives in the house for $30$ years, what is the total amount paid for the house, including down payment, principal, interest, and private mortgage insurance?