Keith received a $30$ year loan of $$225,000$ to purchase a house. The interest rate on the loan was $3\mathrm{.}70\%$ compounded semi$-$annually.

a$.$ What is the size of the monthly loan payment?

Round to the nearest cent

b$.$ What is the balance of the loan at the end of year $4?$

Round to the nearest cent

c$.$ By how much will the amortization period shorten if Keith makes an extra payment of $$30,000$ at the end of year $4?$

years

months