The mortgage on your house is $5$ years old. It required monthly payments of $$1,427\mathrm{.}78,$ had an original term of $30$ years,

and had an interest rate of $8\%($APR$).$ In the intervening $5$ years, interest rates have fallen and so you have decided to

refinance$-$that is$,$ you will roll over the outstanding balance into a new mortgage. The new mortgage has a $30-$year

term, requires monthly payments, and has an interest rate of $5\mathrm{.}625\%($APR$).$

a$.$ What monthly repayments will be required with the new loan?

b$.$ If you still want to pay off the mortgage in $25$ years, what monthly payment should you make after you refinance?

c$.$ Suppose you are willing to continue making monthly payments of $$1,427\mathrm{.}78.$ How long will it take you to pay off the

mortgage after refinancing?

d$.$ Suppose you are willing to continue making monthly payments of $$1,427\mathrm{.}78,$ and want to pay off the mortgage in $25$

years. How much additional cash can you borrow today as part of the refinancing?

$($Note: Be careful not to round any intermediate steps less than six decimal places.$)$