A university research foundation was established by a fund of $$250,000$ invested at a rate of interest

that would provide $$30,000$ payments at the end of each year forever.

$($a$)$ Calculate the rate of interest being earned on the fund. $(3$ marks$)$

$($b$)$ After the payment in $5$ years after establishment, the foundation learned that the rate of

interest on the fund was being changed to $9\%$ per annum. If the foundation wants to continue

annual payments forever, calculate the size of the new payments. $(3$ marks$)$

$($c$)$ If$,$ instead, the foundation continues with the $$30,000$ payments annually, how many full

payments can be made at the new $9\%$ per annum rate of interest?