\( \left.\begin{array}{l}\text { Suppose that you invest } \$ 1000 \text { in a bank account that } \\ \text { pays } 6 \% \text { per year interest APR (annual percentage } \\ \text { rate). In this problem you will explore ways of } \\ \text { predicting how much money you will have in the } \\ \text { account as a function of time. }\end{array} \right\rvert\, \begin{array}{l}\text { 1. How much interest will you have earned by the end } \\ \text { of the first year? How much money, total, will be } \\ \text { in the account at that time? }\end{array} \)

To determine how much interest you will have earned by the end of the first year and the total amount in your account, we can use the **simple interest** formula. Here's a step-by-step breakdown: ### **1. Calculating the Interest Earned** **Simple Interest Formula:** \[ \text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \] Where: - **Principal (\$P)** = \$1,000 - **Rate (\$r$)** = 6% per year = 0.06 (in decimal) - **Time (\$t$)** = 1 year **Plugging in the values:** \[ \text{Interest} = \$1,000 \times 0.06 \times 1 = \$60 \] So, **you will earn \$60 in interest** by the end of the first year. ### **2. Calculating the Total Amount in the Account** **Total Amount Formula:** \[ \text{Total Amount} = \text{Principal} + \text{Interest} \] **Using the calculated interest:** \[ \text{Total Amount} = \$1,000 + \$60 = \$1,060 \] Therefore, **the total amount in your account at the end of the first year will be \$1,060**. ### **Summary** - **Interest Earned:** \$60 - **Total Amount in Account:** \$1,060 This calculation assumes **simple interest**, where interest is calculated only on the principal amount each year. If the interest were **compounded**, the total amount might differ slightly depending on the compounding frequency (e.g., monthly, quarterly).