A person donated $$160,000$ to a university to establish a scholarship fund. What is unusual is that the entire amount came from what the person was able

to save each month over $82$ years while working at a low$-$wage job.

$($a$)$ How much would the person have to put into a savings account at the end of every $3$ months to accumulate $$160,000$ over $82$ years? Assume an

interest rate of $5\mathrm{.}25\%$ compounded quarterly.

$($b$)$ Answer part $($a$)$ using a $2\%$ and a $8\%$ interest rate.

$($a$)$ The necessary deposit is $ at $5\mathrm{.}25\%.$

$($Do not round until the final answer. Then round to the nearest cent as needed.$)$