This is a hypothetical situation. Michelle is $46$ years old and is just now thinking about saving for retirement, which she is planning on at $65.$ Right now she works for the state department, she has been there $1$ year so far, and is expecting a small pension when she retires. The pension amount will be $1\mathrm{.}8\%$ of her average salary multiplied by the number of years she works for the state department.

She does not know a lot about investing but is open to the idea. She is coming to you today to open her retirement account and relying en your experise to navigate ner way to the best outcome for her. She is thinking she will rised to keep her monthly income the same when she retires due to her choices in life so far. SHe also knows that since she has many more years before she retires, you have told her to expect inflation to pay a part of her investment. You tell her that the investment inflation rate is currently $4\%.$

She has been married for $3$ years and they decided to keep their finances separate since they got married late and were "set in their ways." They currently live in an apartment and have a blended family of $2$ kids under the age of $10.$ She has a gross yearly income of between $$76,000-$$$83,000.$ She has a small lump sum of $$3500$ to invest today and she has $$75$ left in her budget at the end of every year to put towards this investment with you.

$1.$ In order to reach her retirement goal, how much does she need in the account at the end of the investment?