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 the sum of an infinite geometric series:** $$ S = \frac{a}{1 - r} $$ where $a$ is the first term and $r$ is the common ratio with $|r| < 1$. 4. **Important note:** The ball travels up and down for each bounce except the first drop. So the total height is the initial height plus twice the sum of the heights of all subsequent bounces. 5. **Calculate the total height:** - Initial height (first bounce up): $3$ m - Common ratio: $r = 0.9$ - Heights of subsequent bounces: $3 \times 0.9, 3 \times 0.9^2, 3 \times 0.9^3, ...$ 6. **Sum of subsequent bounces:** $$ S = \frac{3 \times 0.9}{1 - 0.9} = \frac{2.7}{0.1} = 27 $$ 7. **Total height covered:** - Initial drop: $3$ m - Up and down for subsequent bounces: $2 \times 27 = 54$ m $$ \text{Total height} = 3 + 54 = 57 \text{ meters} $$ **Final answer:** The ball covers a total height of $57$ meters before coming to rest.