1. **Problem statement:** A tennis ball is dropped from a height $h_0=10$ m. Each bounce reaches 80% of the previous height. The time for a full bounce (up and down) from height $h$ is given by $$t = 2\sqrt{\frac{2h}{g}}$$ with $g=10$ m/s$^2$. We want to find how many full bounces occur within 20 seconds. 2. **Formula and explanation:** The height after the $n$-th bounce is $$h_n = h_0 \times 0.8^n$$. The time for the $n$-th bounce is $$t_n = 2\sqrt{\frac{2h_n}{g}} = 2\sqrt{\frac{2h_0 \times 0.8^n}{10}} = 2\sqrt{\frac{2 \times 10 \times 0.8^n}{10}} = 2\sqrt{2 \times 0.8^n}$$. 3. **Simplify the time expression:** $$t_n = 2\sqrt{2} \times (0.8)^{n/2}$$. 4. **Total time for $N$ bounces:** $$T_N = \sum_{n=0}^{N-1} t_n = 2\sqrt{2} \sum_{n=0}^{N-1} (0.8)^{n/2}$$. 5. **Sum of geometric series:** The sum is $$S_N = \sum_{n=0}^{N-1} r^n$$ with $$r = (0.8)^{1/2} = \sqrt{0.8}$$. So, $$S_N = \frac{1 - r^N}{1 - r} = \frac{1 - (\sqrt{0.8})^N}{1 - \sqrt{0.8}}$$. 6. **Total time expression:** $$T_N = 2\sqrt{2} \times \frac{1 - (\sqrt{0.8})^N}{1 - \sqrt{0.8}}$$. 7. **Find maximum $N$ such that $T_N \leq 20$ seconds:** $$2\sqrt{2} \times \frac{1 - (\sqrt{0.8})^N}{1 - \sqrt{0.8}} \leq 20$$ 8. **Isolate the term:** $$\frac{1 - (\sqrt{0.8})^N}{1 - \sqrt{0.8}} \leq \frac{20}{2\sqrt{2}} = \frac{20}{2.8284} \approx 7.0711$$ 9. **Multiply both sides:** $$1 - (\sqrt{0.8})^N \leq 7.0711 (1 - \sqrt{0.8})$$ 10. **Calculate denominator:** $$1 - \sqrt{0.8} = 1 - 0.8944 = 0.1056$$ 11. **Right side:** $$7.0711 \times 0.1056 \approx 0.7467$$ 12. **Inequality:** $$1 - (\sqrt{0.8})^N \leq 0.7467 \implies (\sqrt{0.8})^N \geq 1 - 0.7467 = 0.2533$$ 13. **Take natural logarithm:** $$N \ln(\sqrt{0.8}) \geq \ln(0.2533)$$ 14. **Calculate logs:** $$\ln(\sqrt{0.8}) = \frac{1}{2} \ln(0.8) = \frac{1}{2} (-0.2231) = -0.11155$$ $$\ln(0.2533) = -1.372$$ 15. **Solve for $N$:** $$N \times (-0.11155) \geq -1.372 \implies N \leq \frac{-1.372}{-0.11155} = 12.3$$ 16. **Interpretation:** The maximum integer number of full bounces within 20 seconds is $$\boxed{12}$$. Thus, the ball completes 12 full bounces within 20 seconds.