Hassan Mustafa is going to start a new job as a financial manager in a firm called ScanSoft. To travel to his new job, Hassan is shopping for a new vehicle, and has noticed that many vehicle manufacturers are offering special deals to sell off the current year's vehicles before the models arrive. Hassan's local Ford dealership is advertising $3\mathrm{.}9\%$ financing for a full $48$ months $($i$.$e$.,3\mathrm{.}9\%$ compounded monthly$)$ or up to $$4000$ cash back on selected vehicles.

The vehicle that Hassan wants to purchase costs $$24,600$ including taxes, delivery, licence, and dealer preparation. This vehicle qualifies for $$1800$ cash back if Hassan pays cash for the vehicle. Hassan has a good credit rating and knows that he could arrange a vehicle loan at his bank for the full price of any vehicle he chooses. His other option is to take the dealer financing offered at $3\mathrm{.}9\%$ for $48$ months.

Hassan wants to know which option requires the lower monthly payment.

$1.$ Suppose Hassan buys the vehicle on January $1.$ What monthly payment must Hassan make if he chooses the dealer's $3\mathrm{.}9\%$ financing option and pays off the loan over $48$ months? $($Assume he makes each monthly payment at the end of the month and his first payment is due on January $31.)$

$2.$ Suppose the bank offers Hassan a $48-$month loan with the interest compounded monthly and the payments due at the end of each month. If Hassan accepts the bank loan, he can get $$1800$ cash back on this vehicle.

Help Hassan work out a method to calculate the bank rate of interest required to make bank financing the same cost as dealer financing. First, calculate the monthly rate of interest that would make the monthly bank payments equal to the monthly dealer payments. Then calculate the effective rate of interest represented by the monthly compounded rate. If the financing from the bank is at a lower rate of interest compounded monthly, choose the bank financing. The reason is that the monthly payments for the bank's financing would be lower than the monthly payments for the dealer's $3\mathrm{.}9\%$ financing.

a$.$ How much money would Hassan have to borrow from the bank to pay cash for this vehicle?

b$.$ Using the method above, calculate the effective annual rate of interest and the nominal annual rate of interest required to make the monthly payments for bank financing exactly the same as for dealer financing.

$3.$ Suppose Hassan decides to explore the costs of financing a more expensive vehicle. The more expensive vehicle costs $$34,900$ in total and qualifies for the $3\mathrm{.}9\%$ dealer financing for $48$ months or $$2500$ cash back. What is the highest effective annual rate of interest at which Hassan should borrow from the bank instead of using the dealer's $3\mathrm{.}9\%$ financing?$($answer the question in BA$+$ calculator format avoid using formulas$)$