At the age of 27 , to save for retirement, you decide to deposit \( \$ 70 \) at the end of each month in an IRA that pays \( 4 \% \) compounded monthly. a. Use the following formula to determine how much you will have in the IRA when you retire at age 65 . \[ A=\frac{P\left[(1+r)^{t}-1\right]}{r} \text { or } \] \[ A=\frac{P\left[\left(1+\frac{r}{n}\right)^{n t}-1\right]}{\left(\frac{r}{n}\right)} \] b. Find the interest.

To determine how much you'll have in the IRA when you retire at age 65 using the given formula, let's plug in the numbers: 1. **Monthly Deposit (P):** $70 2. **Annual Interest Rate (r):** 4% or 0.04 3. **Months in a Year (n):** 12 4. **Total Investment Duration (t):** 65 - 27 = 38 years Now we can calculate \( A \): \[ A = \frac{70\left[\left(1+\frac{0.04}{12}\right)^{12 \times 38}-1\right]}{\left(\frac{0.04}{12}\right)} \] Calculating the components: - \( \frac{0.04}{12} = 0.003333 \) - \( 12 \times 38 = 456 \) - \( 1 + 0.003333 = 1.003333 \) - \( \left(1.003333\right)^{456} \approx 4.0688 \) Putting this all together: \[ A \approx \frac{70\left[4.0688-1\right]}{0.003333} \approx \frac{70 \times 3.0688}{0.003333} \approx \frac{214.816}{0.003333} \approx 64,445.15 \] So, you'll have about **$64,445.15** in the IRA when you retire at age 65. To find the interest earned, subtract the total contributions from the total amount in the IRA: - **Total Contributions:** \( 70 \times 12 \times 38 = 70 \times 456 = 31,920 \) - **Interest Earned:** \( 64,445.15 - 31,920 = 32,525.15 \) Therefore, the interest earned will be approximately **$32,525.15**.