1. **State the problem:** We are given the height function of a model rocket as $$h = 100t - 16t^2$$ where $h$ is the height in feet and $t$ is the time in seconds. We want to understand the rocket's height at any time $t$. 2. **Formula and explanation:** The height function is a quadratic equation in terms of $t$. It models the rocket's height as it moves upward and then comes down due to gravity. The term $100t$ represents the initial upward velocity, and $-16t^2$ represents the effect of gravity pulling the rocket down. 3. **Find when the rocket hits the ground:** The rocket hits the ground when $h=0$. Set the equation to zero: $$0 = 100t - 16t^2$$ 4. **Factor the equation:** $$0 = t(100 - 16t)$$ 5. **Solve for $t$:** $$t = 0 \quad \text{or} \quad 100 - 16t = 0$$ 6. **Solve the second equation:** $$16t = 100$$ $$t = \frac{100}{16} = \frac{25}{4} = 6.25$$ 7. **Interpretation:** The rocket is on the ground at $t=0$ seconds (launch time) and again at $t=6.25$ seconds (when it lands). 8. **Find the maximum height:** The maximum height occurs at the vertex of the parabola. The time at vertex is given by: $$t = -\frac{b}{2a} = -\frac{100}{2 \times (-16)} = \frac{100}{32} = 3.125$$ 9. **Calculate maximum height:** Substitute $t=3.125$ into the height equation: $$h = 100(3.125) - 16(3.125)^2 = 312.5 - 16 \times 9.765625 = 312.5 - 156.25 = 156.25$$ 10. **Conclusion:** The rocket reaches a maximum height of 156.25 feet at 3.125 seconds after launch. This explains the rocket's height as a function of time and key points on its trajectory.