Question 2 (1 point) Find " \( m \) " for the investment of \( \$ 1000.00 \) for 2 years at \( 1: 8 \% \) compounde semi-annually.

To find the amount \( m \) for an investment of \( \$1000.00 \) for 2 years at a \( 8\% \) interest rate compounded semi-annually, we can use the formula for compound interest: \[ m = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( m \) = the amount of money accumulated after n years, including interest. - \( P \) = the principal amount (the initial amount of money). - \( r \) = the annual interest rate (decimal). - \( n \) = the number of times that interest is compounded per year. - \( t \) = the number of years the money is invested or borrowed. ### Given: - \( P = 1000 \) - \( r = 0.08 \) (which is \( 8\% \) expressed as a decimal) - \( n = 2 \) (since the interest is compounded semi-annually) - \( t = 2 \) ### Step 1: Substitute the values into the formula Now, we can substitute the known values into the formula: \[ m = 1000 \left(1 + \frac{0.08}{2}\right)^{2 \cdot 2} \] ### Step 2: Calculate the expression inside the parentheses First, calculate \( \frac{0.08}{2} \): \[ \frac{0.08}{2} = 0.04 \] Now substitute this back into the formula: \[ m = 1000 \left(1 + 0.04\right)^{4} \] ### Step 3: Simplify further Now, simplify \( 1 + 0.04 \): \[ 1 + 0.04 = 1.04 \] So now we have: \[ m = 1000 \left(1.04\right)^{4} \] ### Step 4: Calculate \( (1.04)^{4} \) Now we need to calculate \( (1.04)^{4} \): \[ (1.04)^{4} \] ### Step 5: Calculate the final amount \( m \) Now we can calculate \( m \): \[ m = 1000 \cdot (1.04)^{4} \] Let's perform the calculations. Calculate the value by following steps: - step0: Calculate: \(1.04^{4}\) - step1: Convert the expressions: \(\left(\frac{26}{25}\right)^{4}\) - step2: Use the properties of exponents: \(\frac{26^{4}}{25^{4}}\) Calculate or simplify the expression \( 1000 * 1.16985856 \). Calculate the value by following steps: - step0: Calculate: \(1000\times 1.16985856\) - step1: Multiply the numbers: \(1169.85856\) Now that we have calculated \( (1.04)^{4} \) to be approximately \( 1.16985856 \), we can find the final amount \( m \): \[ m = 1000 \cdot 1.16985856 \approx 1169.86 \] ### Final Result The amount \( m \) after 2 years for an investment of \( \$1000.00 \) at \( 8\% \) compounded semi-annually is approximately: \[ \boxed{1169.86} \]