HW Score: \( 60 \%, 6 \) of 10 points To help pay for new costumes for a play, a theater invests \( \$ 1900 \) in a 42-month \( C D \) paying \( 3.8 \% \) interest compounded monthly. Determine the amount the theater will receive when it cashes in the CD after 42 months. The theater will receive \( \$ \square \) when it cashes in the \( C D \). ? (Round to the nearest cent as needed.)

We are given: - Principal \( P = 1900 \) dollars - Annual interest rate \( r = 3.8\% = 0.038 \) - Time period: 42 months, which is \( t = \frac{42}{12} = 3.5 \) years - Interest is compounded monthly, so the number of compounding periods per year is \( n = 12 \). The compound interest formula is: \[ A = P \left(1+\frac{r}{n}\right)^{n t} \] Since the total number of compounding periods is \( n t = 12 \times 3.5 = 42 \), we can write the formula as: \[ A = 1900 \left(1+\frac{0.038}{12}\right)^{42} \] ### Step 1: Calculate the monthly interest rate \[ \frac{r}{n} = \frac{0.038}{12} \approx 0.00316667 \] ### Step 2: Write the expression \[ A = 1900 \left(1.00316667\right)^{42} \] ### Step 3: Compute the compound factor We need to evaluate \(\left(1.00316667\right)^{42}\). One way to approximate this is by using logarithms: \[ \ln\left(1.00316667\right) \approx 0.00316166 \quad (\text{using } \ln(1+x) \approx x - \frac{x^2}{2}) \] Multiply by \(42\): \[ 42 \times 0.00316166 \approx 0.13273 \] Exponentiate: \[ \left(1.00316667\right)^{42} \approx e^{0.13273} \approx 1.14195 \] ### Step 4: Calculate the final amount \[ A \approx 1900 \times 1.14195 \approx 2169.705 \] Rounding to the nearest cent: \[ A \approx \$2169.71 \] ### Final Answer The theater will receive approximately \( \$2169.71 \) when it cashes in the CD.