You just paid full tuition for upcoming your junior year in college. Your plan is to complete this year, then, for your last year in college $($starting one year from now$)$ you will need to pay that final year of undergrad tuition. THEN, after college, you plan to go through a $2-$year DBA program $($this is a way better plan than law school$).$ Today's undergrad tuition you just paid was $$14,000$ and these undergrad prices are expected to increase $4\%$ each year. Today's DBA price, per year, is $$19,000,$ but we expect this DBA price to increase $7\%$ per year every year. Tuition for each DBA year is due at the start of each year $($so the payments due for your DBA program come $2$ and $3$ years from now$).$ Your uncle offers to put you through school, and he will do so by depositing a single lump sum in an investment account, today, which projects a $11\%($APR$)$ annual return. $(7)$

a$.$ How large must your uncle's deposit be$?$

b$.$ If your uncle were willingly to put the deposit amount in $\text{'}a\text{'},$ but you additionally qualified for a scholarship paying $$5,000$ of your senior$-$year undergraduate costs and $$6,000$ per year $($for each of the two years$)$ of your DBA costs, then how much would be left over in the account after you graduate with your DBA $(4$ years from now$).($These scholarship amounts are set, i$.$e$.,$ they will not grow above these levels due to inflation, etc.$).$ Assume the $11\%($APR$)$ annual return rate continues to hold.