a. use the appropriate formula to determine the periodic deposit. b. how much of the financial goal comes…

Explanation:

Step1: Identify the relevant formula

The formula for the future - value of an ordinary annuity is $F = P\times\frac{(1 + \frac{r}{n})^{nt}-1}{\frac{r}{n}}$, where $F$ is the future value of the annuity, $P$ is the periodic payment, $r$ is the annual interest rate (in decimal form), $n$ is the number of times compounded per year, and $t$ is the number of years. We are given that $F=$180000$, $r = 0.045$ (since $4.5%=0.045$), $n = 12$ (compounded monthly), and $t = 10$ years. We need to solve for $P$.

Step2: Rearrange the formula for $P$

Starting with $F = P\times\frac{(1+\frac{r}{n})^{nt}-1}{\frac{r}{n}}$, we can solve for $P$ as $P=\frac{F\times\frac{r}{n}}{(1 + \frac{r}{n})^{nt}-1}$.

Step3: Substitute the values

Substitute $F = 180000$, $r=0.045$, $n = 12$, and $t = 10$ into the formula for $P$. First, calculate $(1+\frac{r}{n})^{nt}=(1+\frac{0.045}{12})^{12\times10}=(1 + 0.00375)^{120}$. Using a calculator, $(1 + 0.00375)^{120}\approx1.565681$. Then, $\frac{r}{n}=\frac{0.045}{12}=0.00375$. $P=\frac{180000\times0.00375}{1.565681 - 1}=\frac{675}{0.565681}\approx1193.25$.

Step4: Round the result

Rounding up to the nearest dollar, $P = 1194$.

Answer:

$1194$