1. **Problem Statement:** We have data for the height $H$ of a firework rocket at different times $t$. We want to find the vertex coordinates and the quadratic equation modeling the rocket's path, then estimate the height at $t=5.5$ seconds. 2. **Data Points:** $(0,0), (1,19.8), (3,46.2), (5,55), (6,52.8), (8,35.2), (10,0)$. 3. **Step 1: Identify the Vertex** The vertex is the highest point on the parabola. From the data, the maximum height is $55$ meters at $t=5$ seconds. So, vertex $V = (5, 55)$. 4. **Step 2: Form of the Quadratic Equation** The height $H$ as a function of time $t$ can be modeled as: $$H = a(t - h)^2 + k$$ where $(h,k)$ is the vertex. Here, $h=5$, $k=55$, so: $$H = a(t - 5)^2 + 55$$ 5. **Step 3: Find $a$ using a known point** Use point $(0,0)$: $$0 = a(0 - 5)^2 + 55$$ $$0 = 25a + 55$$ $$25a = -55$$ $$a = -\frac{55}{25} = -2.2$$ 6. **Step 4: Write the equation** $$H = -2.2(t - 5)^2 + 55$$ 7. **Step 5: Estimate height at $t=5.5$ seconds** Substitute $t=5.5$: $$H = -2.2(5.5 - 5)^2 + 55 = -2.2(0.5)^2 + 55 = -2.2 \times 0.25 + 55 = -0.55 + 55 = 54.45$$ **Final answer:** The rocket's height at $5.5$ seconds is approximately $54.45$ meters.