A college professor$($Prof$.$ M$)$ has two young children. Two years$$ ago, Prof. M invested $$$$180$ comma $000$ in a fund hoping to have $$$$450$ comma $000$ available $11$ years later when his first child started college.$$ However, the$$ account's balance is now only $$$$160$ comma $000.$

a$.$ What was the original annual rate of return needed to reach Prof.$$ M$\text{'}$s goal when he invested in the fund $2$ years$$ ago?

b$.$ Now with only $$$$160$ comma $000$ in the fund and $9$ years remaining until his first child starts$$ college, what APR would the fund have to earn to reach Prof.$$ M$\text{'}$s goal of$$$$450$ comma $000$ if he adds nothing to the$$ account?

c$.$ Shocked by his experience of the past $2$years$,$ Prof. M now wants to invest in a new fund and is willing to make$$ end$-$of$-$month deposits into the fund so that he has the necessary $$$$450$ comma $000$ in $9$ years. The new fund promises to pay a guaranteed APR of $5\mathrm{.}5$ percent compounded monthly. Prof. M decides to transfer the $$$$160$ comma $000$ to this new fund and make the necessary monthly deposits. How large of a monthly deposit must Prof. M make into this new fund to meet his $$$$450$ comma $000$goal$?$

d$.$ Now Prof. M gets sticker shock from the necessary monthly deposit he has to make into the new fund in the preceding question. He decides to invest the $$$$160$ comma $000$ today and $$$$500$ at the end of each month for the next $9$ years into this fund since that is all he can afford. What APR would the fund have to earn for Prof. M to reach his $$$$450$ comma $000$goal$?$