1. The problem states that Hannah saved amounts of 13, 58, 11, 22, and 11 pounds in the first 5 weeks. 2. We are told the mean amount saved over 6 weeks is 22 pounds. 3. The mean (average) is calculated by dividing the total amount saved by the number of weeks: $$\text{mean} = \frac{\text{total savings}}{6}$$ 4. Let the amount saved in the 6th week be $x$. 5. The total savings over 6 weeks is the sum of the first 5 weeks plus $x$: $$13 + 58 + 11 + 22 + 11 + x = 115 + x$$ 6. Using the mean formula: $$22 = \frac{115 + x}{6}$$ 7. Multiply both sides by 6: $$132 = 115 + x$$ 8. Solve for $x$: $$x = 132 - 115 = 17$$ 9. Therefore, Hannah saved **17** pounds in the 6th week.