An Ph$.$D$.$ graduates at age $30,$ and she gets a job that pays $$100,000$ per year. She wants to invest enough to

fund her own retirement without relying on an employer pension program or Social Security. Her goal is to

have $$3$ million saved for retirement at age $60.$ She is relatively confident that her investments will earn an

average interest rate of $7\%$ per year.

$2\mathrm{.}1.$ Assume that she makes equal annual deposits starting on her ${31}^{st}$ birthday and continuing through her

${60}^{th}$ birthday. How much must she invest each year to meet her goal?

$2\mathrm{.}2.$ Suppose she invests the same amount from part $2\mathrm{.}1$ every year starting on her ${41}^{st}$ birthday. How much

money will she have in the account on her ${60}^{th}$ birthday under this scenario $($NOTE: not today's value!$)?$