If \( \$ 26,000 \) is invested in an account for 20 years. Calculate the total interest earned at the end of 20 years if the interest is: (a) \( 8 \% \) simple interest: \( \$ \) (b) \( 8 \% \) compounded annually: \( \$ \) (c) \( 8 \% \) compounded quarterly: \( \$ \) (d) \( 8 \% \) compounded monthly: \( \$ \) Round your answers to the nearest cent. Question Help: Message instructor

To calculate the total interest earned over 20 years for each scenario, we can apply the respective formulas for simple interest and compound interest. For simple interest: \[ \text{Interest} = P \times r \times t \] where \( P \) is the principal amount (\$26,000), \( r \) is the rate (0.08), and \( t \) is the time (20 years). (a) \[ \text{Interest} = 26000 \times 0.08 \times 20 = \$41,600 \] For compound interest, the formula is: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where \( n \) is the number of times interest is compounded per year. (b) Compounded annually (\( n = 1 \)): \[ A = 26000 \left(1 + \frac{0.08}{1}\right)^{1 \times 20} = 26000 \times (1.08)^{20} \approx 26000 \times 4.6604 \approx \$121,173.16 \] Thus, the interest earned is: \[ \$121,173.16 - \$26,000 = \$95,173.16 \] (c) Compounded quarterly (\( n = 4 \)): \[ A = 26000 \left(1 + \frac{0.08}{4}\right)^{4 \times 20} = 26000 \left(1 + 0.02\right)^{80} = 26000 \times (1.02)^{80} \approx 26000 \times 4.6666 \approx \$121,333.91 \] The interest earned is: \[ \$121,333.91 - \$26,000 = \$95,333.91 \] (d) Compounded monthly (\( n = 12 \)): \[ A = 26000 \left(1 + \frac{0.08}{12}\right)^{12 \times 20} = 26000 \left(1 + \frac{0.08}{12}\right)^{240} = 26000 \times (1.00666667)^{240} \approx 26000 \times 4.6847 \approx \$121,753.80 \] Thus, the interest earned is: \[ \$121,753.80 - \$26,000 = \$95,753.80 \] So, rounding to the nearest cent, we have: (a) Simple Interest: \( \$41,600.00 \) (b) Compounded Annually: \( \$95,173.16 \) (c) Compounded Quarterly: \( \$95,333.91 \) (d) Compounded Monthly: \( \$95,753.80 \)