$($Chapter $4\mathrm{.}3,4\mathrm{.}4$ Recall the Future Value of an Annuity formula $A=P\frac{(1+\frac{r}{n}{)}^{nt}-1}{\frac{r}{n}}$ from the Module $11$ Discussion

Board.

a$)$ Solve the formula for $t$ such that your final answer has a single logarithmic function.

b$)$ Find the time it takes to reach $$750,000$ earning $5\%$ real returns with a $$1,\frac{000}{?}$ month contribution.

Suppose at the beginning of year $7$ of saving for retirement, you have $$173,000$ in your retirement account. You contributed $$2,000$ each month that year and at the end of year $7$ you had $$235,000.$

a$)$ Create an equation to find the rate of return in year $7.$ Hint: You will use both the compound interest formula as well as the future value of an annuity formula.

b$)$ Reduce this formula to a polynomial equation in standard form. Hint: I would substitute $x=1+\frac{r}{n}$ to make things easier.

c$)$ Use technology to solve the equation for the rate of return. Typing the polynomial equation into google will work.

$($Chapter $4\mathrm{.}6)$ Consider the following dataset of the value of a $$1$ invested by age at age $65$ with a dollar$-$for$-$dollar company match.

a$)$ What should the value of the graph at age $65$ be$?$

Wealth Multiplier By Age

b$)$ Without a dollar$-$for$-$dollar match, how would the dollar amounts change?

c$)$ Fit an exponential function to the data that includes the employer match using the first data point and the data point for age $65$ with the value from part $a.$

d$)$ How well do you think your model fits the data? Does the model overestimate or underestimate? Would different points produce a better model?

e$)$ Use your model to interpolate values for your current age and the age you will enter the workforce if those ages are

Including a dollar fordollar amployer match

not in the dataset $(20,25,30,$dots$).$ Otherwise, choose the age $22.$