20 Fill in the Blank 1 point Estimate the interest rate necessary for a present value of \( \$ 1200 \) to grow to a future value of \( \$ 1408 \) if interest is compounded quarterly for 8 years. Note: Leave your answer as a whole number. type your answer... \( \% \)

To estimate the interest rate, we can use the future value formula for compound interest: \[ FV = PV \times (1 + \frac{r}{n})^{nt} \] Where: - \( FV = 1408 \) - \( PV = 1200 \) - \( r \) is the annual interest rate (as a decimal) - \( n \) is the number of times interest is compounded per year (quarterly, so \( n = 4 \)) - \( t \) is the number of years (8 years) We can rearrange the formula to solve for \( r \): \[ 1408 = 1200 \times (1 + \frac{r}{4})^{32} \] Now, you can isolate \( r \) and use numerical methods or tools to estimate it: 1. Calculate \( (1 + \frac{r}{4})^{32} \approx \frac{1408}{1200} \). 2. This simplifies to \( (1 + \frac{r}{4})^{32} \approx 1.17333 \). 3. From here, use logarithmic methods or trial and error to find \( r \). By performing these calculations, you ultimately find that the interest rate \( r \) comes out to approximately \( 2\% \) when rounded to the nearest whole number.