$($a$)$ A student invests $$150,000$ at $9\%$ compound interest to pay

for the courses she will take in the spring semester in three

years$$ time. She expects the total costs for the spring semester in

three years$$ time to be $$205,000.($i$)$ What is the accumulated amount she would have available in

three years$$ time?$[7$ marks$]($ii$)$ Would she be able to cover her semester fees? $[2$

marks$]($iii$)$What if the student borrows the $$205,000$ at $32\%$ simple

interest for $24$ months; How much money would she have to repay at

the end of the $24$ month period?$[6$ marks$]($iv$)$Discuss any other options the student could have tapped into

to secure thesemesters$$fee$.[5$ marks$]$