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 $$4,200$ away each month $($ $$350),$ how much will she have in $47$ years at the $6\%$ 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 $9\%$ 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 per month when she is putting money away at $9\%$ than when she was earning a lower rate of $6\%$ over the $47$ years?

Note: Ignore the effect of taxes on your calculations.

a$.$ If Denise puts away $$4,200$ at the end of each year and earns $6\%$ 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.$)$