a$.$ Suppose that between the ages of $22$ and $38,$ you contribute $$$$1000$ per year to a$401($k$)$ and your employer contributes $$$$500$ per year on your behalf. The interest rate is $7\mathrm{.}5\%$ compounded annually. What is the value of the$401($k$)$ after $16$years$?$ b$.$ Suppose that after $16$ years of working for this$$ firm, you move on to a new job.$$ However, you keep your accumulated retirement funds in the$401($k$).$ How much money will you have in the plan when you reach age$65?$ c$.$ What is the difference between the amount of money you will have accumulated in the$401($k$)$ and the amount you contributed to the$$ plan?

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Part $1$

a$.$ The value of the$401($k$)$ after $16$ years is $$$

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$($Do not round until the final answer. Then round to the nearest dollar as$$ needed.$)$