1. **State the problem:** A rubber ball is dropped from a height of 12 meters. Each bounce reaches $ \frac{2}{3} $ of the previous height. We want to find how many times the ball bounces at least 12 cm (0.12 m) high. 2. **Formula and explanation:** The height after the $n^{th}$ bounce is given by the geometric sequence: $$ h_n = 12 \times \left(\frac{2}{3}\right)^n $$ We want to find the largest integer $n$ such that: $$ h_n \geq 0.12 $$ 3. **Set up the inequality:** $$ 12 \times \left(\frac{2}{3}\right)^n \geq 0.12 $$ 4. **Divide both sides by 12:** $$ \cancel{12} \times \left(\frac{2}{3}\right)^n \geq \frac{0.12}{\cancel{12}} $$ $$ \left(\frac{2}{3}\right)^n \geq 0.01 $$ 5. **Take natural logarithm of both sides:** $$ \ln\left(\left(\frac{2}{3}\right)^n\right) \geq \ln(0.01) $$ $$ n \ln\left(\frac{2}{3}\right) \geq \ln(0.01) $$ 6. **Since $ \ln\left(\frac{2}{3}\right) < 0 $, dividing by it reverses inequality:** $$ n \leq \frac{\ln(0.01)}{\ln\left(\frac{2}{3}\right)} $$ 7. **Calculate values:** $$ \ln(0.01) = -4.60517 $$ $$ \ln\left(\frac{2}{3}\right) = -0.40547 $$ $$ n \leq \frac{-4.60517}{-0.40547} = 11.36 $$ 8. **Interpretation:** The ball bounces at least 0.12 m high for $ n = 1, 2, ..., 11 $ times. **Final answer:** The ball will bounce at least 12 cm high **11 times**.