Question $22.($Retirement Problem$)$ Alice Applegate has just graduated from university and will be starting a new job in the treasury department of a major Canadian corporation. She wants to retire in $40$ years. She expects to live for $30$ years in retirement. She wants to receive $$3,000$ per month during her retirement. She can earn $12\%$ compounded quarterly on her investment account.

$(1)$ How much must she save every month while she works in order to have enough to fund her retirement? $(8$ marks$)$

l$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$l$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$l $04070$

$?/$month $$3,000/$month

$(2).$ In addition the above, Alice wants to make an endowment to her alma mater in the form of a scholarship. Upon her death, she wants to donate enough to provide the university with a scholarship of $$5,000$ every year forever to help business students in financial need. The university will invest her donation in an account earning $6\%$ compounded monthly. How much will she have to save every month now to achieve all her plans and vision? $(5$ marks$)$