1. **State the problem:** We are given the height of a rocket as a function of time: $$h = -2(t+1)(t-9)$$ and need to answer several questions about the rocket's height and timing. 2. **Formula and rules:** This is a quadratic function in factored form. The roots (where height is zero) are found by setting each factor to zero. The vertex (maximum height) occurs at the midpoint of the roots because the parabola opens downward (coefficient of $t^2$ is negative). 3. **Part a) Height of the building at launch (time $t=0$):** $$h = -2(0+1)(0-9) = -2(1)(-9) = 18$$ The building height is 18 meters. 4. **Part b) When does the rocket hit the ground?** Set height to zero: $$0 = -2(t+1)(t-9)$$ So, $$t+1=0 \Rightarrow t=-1$$ $$t-9=0 \Rightarrow t=9$$ The rocket hits the ground at $t=9$ seconds (ignoring negative time). 5. **Part c) Maximum height and time:** The vertex time is the midpoint of roots: $$t = \frac{-1 + 9}{2} = 4$$ Calculate height at $t=4$: $$h = -2(4+1)(4-9) = -2(5)(-5) = 50$$ Maximum height is 50 meters at 4 seconds. 6. **Part d) Height at $t=8$ seconds:** $$h = -2(8+1)(8-9) = -2(9)(-1) = 18$$ Height at 8 seconds is 18 meters. **Summary:** - Building height at launch: 18 meters - Rocket hits ground at 9 seconds - Maximum height: 50 meters at 4 seconds - Height at 8 seconds: 18 meters