W-awy.aleks.com/alekscgi|x|isl.exe/10_u-IgNsikr7j8P3jH-IB-ukBiXnxj_1SoekWMaP1As1fEhnci79H4SSUelERg1SH4zrgehQQhBhWi4ArD9Jmx Consumer Mathemadis \( 0 / 3 \) Finding the pertodic payment needed to meet an investment goal The Morgan family wants to save money to travel the world. They plan to invest in an ordinary annuity that earns \( 4.2 \% \) interest, compounded monthly. Payments will be made at the end of each month. How much money do they need to pay into the annuity each month for the annuity to have a total value of \( \$ 13,000 \) after 11 years? Do not round intermediate computations, and round your final answer to the nearest cent. If necessary, refer to the list of financial formulas. \[ \$ 2 \]

We use the future value formula for an ordinary annuity: \[ FV = PMT \cdot \frac{(1+i)^n - 1}{i} \] where \(FV = 13\,000\) (the desired future value), \(i = \frac{0.042}{12}\) (the monthly interest rate), and \(n = 11 \times 12 = 132\) (the total number of months). First, solve for \(PMT\): \[ PMT = \frac{FV \cdot i}{(1+i)^n - 1} \] Substitute in the given values: \[ i = \frac{0.042}{12} = 0.0035; \] \[ PMT = \frac{13\,000 \times 0.0035}{(1.0035)^{132} - 1} \] Next, compute \((1.0035)^{132}\). Using the property of exponentials: \[ (1.0035)^{132} = e^{132 \ln(1.0035)} \] Estimate \(\ln(1.0035) \approx 0.003493\); therefore: \[ 132 \times 0.003493 \approx 0.4608,\quad \text{and} \quad e^{0.4608} \approx 1.585 \] So, we have: \[ (1.0035)^{132} - 1 \approx 1.585 - 1 = 0.585 \] Now, calculate the numerator: \[ 13\,000 \times 0.0035 = 45.5 \] Thus, the monthly payment is: \[ PMT \approx \frac{45.5}{0.585} \approx 77.78 \] The Morgan family needs to pay approximately \(\$77.78\) each month.