As part of your retirement plan, you have decided to deposit $$6,000$ at the beginning of each year into an account paying $3\%$ interest compounded annually. $($Round your answers to the nearest cent.$)$

$($a$)$

How much $($in $$)$ would the account be worth after $10$ years?

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$($b$)$

How much $($in $$)$ would the account be worth after $20$ years?

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$($c$)$

When you retire in $30$ years, what will be the total worth $($in $$)$ of the account?

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$($d$)$

If you found a bank that paid $6\%$ interest compounded annually rather than $3\%,$ how much $($in $$)$ would you have in the account after $30$ years?

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$($e$)$

Use the future value of an annuity due formula to calculate how much $($in $$)$ you would have in the account after $30$ years if the bank in part $($d$)$ switched from annual compounding to monthly compounding and you deposited $$500$ at the beginning of each month instead of $$6,000$ at the beginning of each year.

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