13. From 2015 to 2022 , hourly wages raised by a fixed percent every year. If the wage was \( \$ 8 / \mathrm{h} \) in 2015 and \( \$ 10.04 / \mathrm{h} \) in 2017 , how much is the hourly rate in 2022 ? What is the percent of increase each year?

To solve the problem of determining the hourly wage in 2022 and the annual percentage increase, we'll follow these steps: 1. **Identify the Given Information:** - **Initial Wage (2015):** \$8/hour - **Wage in 2017:** \$10.04/hour - **Time Span from 2015 to 2022:** 7 years 2. **Determine the Annual Growth Rate:** Since the wages increase by a fixed percentage each year (compound interest), we can use the compound interest formula: \[ \text{Final Wage} = \text{Initial Wage} \times (1 + r)^n \] Where: - \( \text{Final Wage} \) is the wage after \( n \) years. - \( r \) is the annual growth rate (decimal). - \( n \) is the number of years. **From 2015 to 2017:** - **Number of Years (n):** 2 - **Equation:** \[ 10.04 = 8 \times (1 + r)^2 \] - **Solving for \( r \):** \[ (1 + r)^2 = \frac{10.04}{8} = 1.255 \] \[ 1 + r = \sqrt{1.255} \approx 1.12 \] \[ r \approx 0.12 \text{ or } 12\% \] 3. **Calculate the Hourly Wage in 2022:** Now, using the annual growth rate of 12%, we'll calculate the wage after 7 years (from 2015 to 2022). \[ \text{Wage in 2022} = 8 \times (1 + 0.12)^7 \] **Calculating \( (1.12)^7 \):** \[ (1.12)^7 \approx 2.21068 \] \[ \text{Wage in 2022} = 8 \times 2.21068 \approx 17.68544 \] **Rounded to Two Decimal Places:** \[ \text{Wage in 2022} \approx \$17.69/\text{hour} \] 4. **Summary of Results:** - **Annual Percentage Increase:** Approximately **12%** per year. - **Hourly Wage in 2022:** Approximately **\$17.69/hour**. **Final Answer:** - **2022 Hourly Wage:** Approximately \$17.69 per hour - **Annual Increase Rate:** 12 % each year