1. **State the problem:** A ball bounces to a height of 3 meters initially, and each subsequent bounce reaches a height 10% lower than the previous one. We need to find the total height covered by the ball before it comes to rest. 2. **Identify the type of problem:** This is a geometric series problem where each term is 90% (100% - 10%) of the previous term. 3. **Formula for total height:** The total height covered includes the initial height plus twice the sum of all subsequent bounce heights (up and down), except the first drop. 4. **Define variables:** - Initial height $h_1 = 3$ m - Common ratio $r = 0.9$ 5. **Calculate the sum of infinite geometric series:** The sum of infinite terms starting from the second bounce is $$S = \frac{h_1 \times r}{1 - r} = \frac{3 \times 0.9}{1 - 0.9} = \frac{2.7}{0.1} = 27$$ 6. **Calculate total height:** The ball first falls 3 m, then bounces up and down for each subsequent height. So total height is: $$\text{Total height} = h_1 + 2 \times S = 3 + 2 \times 27 = 3 + 54 = 57$$ 7. **Answer:** The total height covered by the ball before coming to rest is **57 meters**.