A $$300,000$ mortgage was initially established with $30$ years of monthly payments at $8\mathrm{.}0\%AR$

interest. After $10$ years $(120$ payments$),$ you observe that market interest rates have decreased to

$4\mathrm{.}0\%$ AR for a new $20$ year mortgage. You are considering refinancing the existing mortgage

with monthly payments over the remaining $20$ years. Refinancing charges amount to $$2,500.$

a$.$ If you maintain the payments at the original level, how much money would the refinancing

make available to remodel a room in your house?

b$.$ Suppose you only needed $$40,000$ to remodel your house. By how much could you lower

your existing payment level if you refinanced an amount sufficient to pay off the existing

mortgage, cover your closing costs, and provide you with $$40,000$ for the remodeling

project?