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.)

To determine the amount the theater will receive when it cashes in the CD after 42 months, we can use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - \(A\) is the amount of money accumulated after n years, including interest. - \(P\) is the principal amount (the initial amount of money). - \(r\) is the annual interest rate (decimal). - \(n\) is the number of times that interest is compounded per year. - \(t\) is the time the money is invested or borrowed (in years). For this problem: - \(P = 1900\) - \(r = 3.8\% = 0.038\) - \(n = 12\) (because the interest is compounded monthly) - \(t = 42 \text{ months} = 42/12 = 3.5 \text{ years}\) Now substituting the values into the formula: \[ A = 1900 \left(1 + \frac{0.038}{12}\right)^{12 \times 3.5} \] \[ A = 1900 \left(1 + \frac{0.038}{12}\right)^{42} \] \[ A = 1900 \left(1 + 0.00316667\right)^{42} \] \[ A = 1900 \left(1.00316667\right)^{42} \] \[ A = 1900 \left(1.143781643\right) \quad(\text{using a calculator}) \] \[ A \approx 2173.19 \] So, the theater will receive approximately \( \$ 2173.19 \) when it cashes in the CD after 42 months.