Future value with periodic rates. Denise has her heart set on being a millionaire. She decides that at the end of every year, she will put away $$4,800$ into her "I want to be a millionaire account" at her local bank. She expects to earn $5\mathrm{.}5\%$ annually on her account.

a$.$ How many years must Denise faithfully put away her money to succeed at becoming a millionaire?

b$.$ If Denise switches to a monthly savings plan and puts one$-$twelfth of the $$4,800$ away each month $($ $$400),$ how much will she have in $47$ years at the $5\mathrm{.}5\%$ APR?

c$.$ Why is the future value under the monthly savings plan more than the $$1,000,000$ goal?

Next, let's assume Denise is now thirty$-$five years old and thus has thirty years for saving toward her one$-$million$-$dollar goal. She anticipates an APR of $8\mathrm{.}5\%$ on her investments.

d$.$ How much does she need to save each year to become a millionaire by age sixty$-$five if she puts money away annually?

e$.$ How much does she need to save if she puts money away monthly?

f$.$ Why does it take more fier month when she is putting money away at $8\mathrm{.}5\%$ than when she was earning a lower rate of $5\mathrm{.}5\%$ over the $47$ years?

Note: Ignore the effect of taxes on your calculations.

a$.$ If Denise puts away $$4,800$ at the end of each year and earns $5\mathrm{.}5\%$ annually on her account, how many years must she faithfully put away her money to succeed at becoming a millionaire?

years $($Round to two decimal places.$)$