1. **State the problem:** We are given the height of a football as a function of time: $$h = -9.8t^2 + 52.92t + 2.2$$ and need to find when the football reaches its maximum height and what that maximum height is. 2. **Formula and method:** Since the equation is quadratic with a negative leading coefficient, the graph is a downward-opening parabola. The maximum height occurs at the vertex. We use completing the square to find the vertex form. 3. **Rewrite the equation:** Factor out the coefficient of $t^2$ from the first two terms: $$h = -9.8\left(t^2 - \frac{52.92}{9.8}t\right) + 2.2 = -9.8\left(t^2 - 5.4t\right) + 2.2$$ 4. **Complete the square:** Take half of the coefficient of $t$, which is $-5.4$, half is $-2.7$, square it: $$(-2.7)^2 = 7.29$$ Add and subtract inside the parentheses: $$h = -9.8\left(t^2 - 5.4t + 7.29 - 7.29\right) + 2.2$$ 5. **Rewrite as a perfect square:** $$h = -9.8\left((t - 2.7)^2 - 7.29\right) + 2.2$$ 6. **Distribute:** $$h = -9.8(t - 2.7)^2 + 9.8 \times 7.29 + 2.2$$ Calculate $9.8 \times 7.29$: $$9.8 \times 7.29 = 71.442$$ So, $$h = -9.8(t - 2.7)^2 + 71.442 + 2.2 = -9.8(t - 2.7)^2 + 73.642$$ 7. **Interpretation:** The vertex is at $t = 2.7$ seconds, which is when the football reaches maximum height. 8. **Maximum height:** The maximum height is the constant term in vertex form: $$h_{max} = 73.64$$ meters (rounded to 2 decimal places). **Final answers:** - The football reaches maximum height at $t = 2.7$ seconds. - The maximum height is $73.64$ meters.