To determine the range of the given data set, we first need to understand what the range is. The range of a set of numbers is defined as the difference between the maximum and minimum values in that set. ### Step-by-Step Explanation 1. **Identify the Data Points**: The data points provided are: - \( k + 2 \) - \( k - 3 \) - \( k + 4 \) - \( k - 2 \) - \( k \) - \( k - 5 \) - \( k + 3 \) - \( k - 1 \) - \( k + 6 \) 2. **Find the Maximum Value**: To find the maximum value, we need to evaluate each expression: - \( k + 2 \) - \( k - 3 \) - \( k + 4 \) - \( k - 2 \) - \( k \) - \( k - 5 \) - \( k + 3 \) - \( k - 1 \) - \( k + 6 \) Among these, the term \( k + 6 \) is clearly the largest because it adds the highest constant (6) to \( k \). 3. **Find the Minimum Value**: Next, we evaluate the minimum value: - \( k + 2 \) - \( k - 3 \) - \( k + 4 \) - \( k - 2 \) - \( k \) - \( k - 5 \) - \( k + 3 \) - \( k - 1 \) - \( k + 6 \) Here, the term \( k - 5 \) is the smallest because it subtracts the highest constant (5) from \( k \). 4. **Calculate the Range**: Now that we have both the maximum and minimum values, we can calculate the range: \[ \text{Range} = \text{Maximum} - \text{Minimum} = (k + 6) - (k - 5) \] Simplifying this: \[ \text{Range} = k + 6 - k + 5 = 11 \] ### Conclusion The range of the data set is **11**. Therefore, the correct option is **B**. ### Explanation of Other Options - **Option A (10)**: This option is incorrect because it does not account for the correct maximum and minimum values derived from the data set. - **Option C (6)**: This option is also incorrect as it underestimates the difference between the maximum and minimum values. - **Option D (8)**: This option is incorrect for the same reason as option C; it does not reflect the actual difference calculated. ### Revision Summary - The range is calculated as the difference between the maximum and minimum values in a data set. - The maximum value from the data set is \( k + 6 \) and the minimum value is \( k - 5 \). - The range is calculated as \( (k + 6) - (k - 5) = 11 \). - The correct answer is **B (11)**, while the other options are incorrect due to miscalculations of the range.