1. **State the problem:** We want to find the time intervals when the rocket's height $h$ is above 40 meters. 2. **Given model:** The height at time $t$ is given by $$h = -5t^2 + 20t + 25$$ 3. **Set up the inequality:** We want to find $t$ such that $$-5t^2 + 20t + 25 > 40$$ 4. **Simplify the inequality:** Subtract 40 from both sides: $$-5t^2 + 20t + 25 - 40 > 0$$ $$-5t^2 + 20t - 15 > 0$$ 5. **Divide the inequality by -5:** Remember to reverse the inequality sign when dividing by a negative number: $$t^2 - 4t + 3 < 0$$ 6. **Factor the quadratic:** $$t^2 - 4t + 3 = (t - 1)(t - 3)$$ 7. **Analyze the inequality:** $$(t - 1)(t - 3) < 0$$ means the product is negative, so $t$ is between the roots: $$1 < t < 3$$ 8. **Interpretation:** The rocket is above 40 meters between 1 second and 3 seconds after launch. **Final answer:** The rocket is above 40 meters for $$t \in (1, 3)$$ seconds.