Sheldon Cooper plans to buy a research facility for $$350,000.$ He will borrow the

entire amount. Sheldon will take out an adjustable rate mortgage that has an APR

of $3\mathrm{.}0$ percent rate for the first $5$ years. After that the rate will change annually to

reflect market conditions. The annual cap is $2\%($i$.$e$.,$ the largest increase in any year

is $2\%).$ The loan term is $30$ years and payments are made monthly.

What is the remaining balance on the loan after five years?

What is the greatest mortgage payment Sheldon might owe when the interest rate

resets at the beginning of the $6$th year?