Detarmine the following estimate without using a calculator. Then use a calculator to pefform the computation necassary to obtain an exact answer. How reasonable is your estimate when comparsd to the actual answer? If a person who works 40 hours per week eams \( \$ 89,300 \) per year, estimate that persor's hourly wage by rounding the annual income to the nearest ten thousand. Use the estimate that there are 50 weeks in a year. The estimate of that persor's hourly wage is approximately \( \$ 35 \) \( \square \) 35. (Round to the nearest dollar as needed.) Use 52 weeks a year to find the actual hourly wage. The achual hourly wage is \( \$ \) \( \square \) (Round to the nearest cent as needed.)

Let's walk through the problem step by step to determine both the estimated and actual hourly wages. ### **1. Estimating the Hourly Wage Without a Calculator** **a. Round the Annual Income:** - **Original Annual Income:** \$89,300 - **Rounded to the Nearest Ten Thousand:** \$90,000 **b. Calculate Total Estimated Hours Worked in a Year:** - **Hours per Week:** 40 - **Estimated Weeks per Year:** 50 - **Total Hours:** \( 40 \text{ hours/week} \times 50 \text{ weeks} = 2,000 \text{ hours} \) **c. Compute the Estimated Hourly Wage:** \[ \text{Estimated Hourly Wage} = \frac{\text{Rounded Annual Income}}{\text{Total Estimated Hours}} = \frac{\$90,000}{2,000 \text{ hours}} = \$45 \text{ per hour} \] **\(\square\) Filled Estimate:** \[ \text{The estimate of that person's hourly wage is approximately } \$45. \] *Note:* It appears there might have been a typographical error in the initial estimate of \$35. Based on the calculations, \$45 is the accurate estimate. ### **2. Calculating the Actual Hourly Wage Using a Calculator** **a. Calculate Total Actual Hours Worked in a Year:** - **Hours per Week:** 40 - **Actual Weeks per Year:** 52 - **Total Hours:** \( 40 \text{ hours/week} \times 52 \text{ weeks} = 2,080 \text{ hours} \) **b. Compute the Actual Hourly Wage:** \[ \text{Actual Hourly Wage} = \frac{\text{Annual Income}}{\text{Total Actual Hours}} = \frac{\$89,300}{2,080 \text{ hours}} \approx \$42.93 \text{ per hour} \] **\(\square\) Filled Actual Wage:** \[ \text{The actual hourly wage is } \$42.93. \] ### **3. Comparing the Estimate to the Actual Wage** - **Estimated Hourly Wage:** \$45.00 - **Actual Hourly Wage:** \$42.93 **Reasonableness of the Estimate:** - The estimated hourly wage of \$45 is **slightly higher** than the actual wage of \$42.93. - **Difference:** \$45.00 - \$42.93 = \$2.07 - **Percentage Difference:** \( \frac{\$2.07}{\$42.93} \times 100 \approx 4.82\% \) A **4.82%** difference indicates that the estimate is quite reasonable and closely aligns with the actual hourly wage. ### **Summary** - **Estimated Hourly Wage:** \$45 - **Actual Hourly Wage:** \$42.93 - **Reasonableness:** The estimate is very close to the actual wage, differing by less than 5%, which makes it a reasonable approximation.