Question: Jerod hopes to earn $1200 in interest in 4.3 years time from $48,000 that he has available to…

Jerod hopes to earn $1200 in interest in 4.3 years time from $48,000 that he has available to invest. To decide if it’s feasible to do this by investing in an account that compounds semi-annually, he needs to determine the annual interest rate such an account would have to offer for him to meet his goal. What would the annual rate of interest have to be? Round to two decimal places.

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Solution

Jerod wants to earn $1200 in interest in 4.3 years from an investment of $48,000. The account compounds interest semi-annually. We need to determine the annual interest rate. We’ll use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - \(A\) is the future value of the investment, including interest. - \(P\) is the principal investment amount ($48,000). - \(r\) is the annual interest rate (decimal). - \(n\) is the number of times the interest is compounded per year (2 for semi-annual). - \(t\) is the time in years (4.3). First, find \(A\): \[ A = P + \text{Interest} = 48000 + 1200 = 49200 \] Substitute the known values into the compound interest formula: \[ 49200 = 48000 \left(1 + \frac{r}{2}\right)^{2 \times 4.3} \] Simplify the exponent: \[ 49200 = 48000 \left(1 + \frac{r}{2}\right)^{8.6} \] Divide both sides by 48000 to isolate the exponential term: \[ 1.025 = \left(1 + \frac{r}{2}\right)^{8.6} \] Take the 8.6th root of both sides to solve for \(1 + \frac{r}{2}\): \[ 1 + \frac{r}{2} = 1.025^{\frac{1}{8.6}} \] Calculate the 8.6th root: \[ 1 + \frac{r}{2} \approx 1.002802 \] Subtract 1 from both sides: \[ \frac{r}{2} \approx 0.002802 \] Multiply both sides by 2 to solve for \(r\): \[ r \approx 0.005604 \] Convert the decimal to a percentage: \[ r \approx 0.5604\% \] Rounding to two decimal places, the annual rate of interest is approximately 0.56%.