File Edit Format View Help Amishi deposited \( \$ 800 \), at the end of each six months for 18 years in a savings account. If the account paid \( 8 \% \) interest, compounded semiannually, use the appropriate formula to find the future value of | her account. (Round your answer to the nearest cent.)

We begin by recognizing that this is an annuity problem where Amishi makes regular deposits at the end of each period. The future value \( FV \) of an annuity (with deposits at the end of each period) is given by \[ FV = R \cdot \frac{(1 + i)^n - 1}{i} \] where - \( R \) is the periodic deposit, - \( i \) is the interest rate per period, and - \( n \) is the total number of deposits (periods). **Step 1. Determine the parameters** - The deposit per period is \( R = 800 \). - The annual interest rate is \( 8\% \). Since interest is compounded semiannually, the rate per six months is \[ i = \frac{8\%}{2} = 4\% = 0.04. \] - Amishi makes deposits for \( 18 \) years, and since there are 2 periods per year, the total number of periods is \[ n = 18 \times 2 = 36. \] **Step 2. Substitute into the formula** \[ FV = 800 \cdot \frac{(1 + 0.04)^{36} - 1}{0.04} \] **Step 3. Calculate \( (1 + i)^n \)** \[ (1.04)^{36} \] This can be computed using a calculator to obtain a numerical value. Using the exponential method: \[ (1.04)^{36} \approx \exp(36 \ln 1.04) \approx \exp(36 \times 0.03922) \approx \exp(1.4119) \approx 4.104. \] **Step 4. Complete the future value calculation** Substitute this value into the annuity formula: \[ FV = 800 \cdot \frac{4.104 - 1}{0.04} \] \[ FV = 800 \cdot \frac{3.104}{0.04} \] \[ FV = 800 \cdot 77.60 \approx 62,\!080. \] Thus, the future value of her account is approximately \(\$62,\!080.00\).