$($Solving for a comprehensive problem$)$ Suppose that you are in the fall of your senior year and are faced with the choice of either getting a job when you graduate or going to law school. Of course, your choice is not purely financial. However, to make an informed decision, you would like to know the financial implications of the two alternatives. Let's assume that your alternatives are as follows:

If you take the "get a job" route, you expect to start off with a salary of $$45,000$ per year. There is no way to predict what will happen in the future, but your best guess is that your salary will grow at $3$ percent per year until you retire in $41$ years.

As a law student, you will be paying $$20,000$ per year tuition for each of the $3$ years you are in graduate school. However, you can then expect a job with a starting salary of $$75,000$ per year. Moreover, you expect your salary to grow by $8$ percent per year until you retire $35$ years later.

Clearly, your total expected lifetime salary will be higher if you become a lawyer. However, the additional future salary is not free. You will be paying $$20,000$ in tuition at the beginning of each of the $3$ years of law school. In addition, you will be giving up a little more than $$139,000$ in lost income over the $3$ years of law school: $$45,000$ the first year, $$46,350$ the second year, and $$47,741$ the third year.

a$.$ To start your analysis of whether to go to law school, calculate the present value of the future earnings that you will realize by going directly to work, assume a discount rate of $10$ percent.

b$.$ What is the present value today of your future earnings if you decide to attend law school, assuming a discount rate of $10$ percent? Remember that you will be in law school for $3$ years before you start to work as a lawyer. $($Hint: assume that you are paid at the end of each year, so that your first salary payment if you decide to go to law school occurs $4$ years from now.$)$

c$.$ If you pay your law school tuition at the beginning of each year, what is the present value of your tuition, assuming a discount rate of $10$ percent?

a$.$ What is the present value of the future earnings that you will realize by going directly to work, assuming a discount rate of $10$ percent?

$ $($Round to the nearest cent.$)$