Josie borrows 2000 and repays a final amount of 2186.17 in 2 years. What annual pre-of interest, compounded semi-annually, is she being offered on this loan? A) 4% B) 3% C) 4.5% D) 3.5% Question 3(1 point)

## Step 1: Convert the annual interest rate to a semi-annual rate.<br />### Let $r$ be the annual interest rate. Since the interest is compounded semi-annually, the semi-annual interest rate is $\frac{r}{2}$.<br /><br />## Step 2: Calculate the number of compounding periods.<br />### The loan is for 2 years, and the interest is compounded semi-annually (twice a year). Therefore, the number of compounding periods is $2 \times 2 = 4$.<br /><br />## Step 3: Set up the compound interest formula.<br />### The formula for compound interest is $A = P(1 + \frac{r}{n})^{nt}$, where $A$ is the final amount, $P$ is the principal, $r$ is the annual interest rate, $n$ is the number of times interest is compounded per year, and $t$ is the time in years. In this case, $A = \$2186.17$, $P = \$2000$, $n = 2$, and $t = 2$. We need to solve for $r$. So, $2186.17 = 2000(1 + \frac{r}{2})^{2 \times 2}$.<br /><br />## Step 4: Solve for the annual interest rate.<br />### Simplify the equation: $2186.17 = 2000(1 + \frac{r}{2})^4$.<br />### Divide both sides by 2000: $\frac{2186.17}{2000} = (1 + \frac{r}{2})^4$.<br />### $1.093085 = (1 + \frac{r}{2})^4$<br />### Take the fourth root of both sides: $\sqrt[4]{1.093085} = 1 + \frac{r}{2}$.<br />### $1.0225 = 1 + \frac{r}{2}$<br />### Subtract 1 from both sides: $0.0225 = \frac{r}{2}$<br />### Multiply both sides by 2: $r = 0.045$<br />### Convert to percentage: $r = 4.5\%$