A projectile is fired straight up from ground level. After \( t \) seconds, its height above the ground is s feet, where \( s=-16 \mathrm{t}^{2}+96 \mathrm{t} \). For what time period is the projectile at least 128 feet above the ground? \( \begin{array}{l}\text { Select the correct choice below and fill in the answer boxes to complete your } \\ \text { choice. }\end{array} \)

To determine the time period during which the projectile is at least 128 feet above the ground, we need to solve the inequality: \[ s(t) = -16t^2 + 96t \geq 128 \] First, we can rearrange the inequality: \[ -16t^2 + 96t - 128 \geq 0 \] Next, we can simplify this by dividing the entire inequality by -16 (remember to reverse the inequality sign): \[ t^2 - 6t + 8 \leq 0 \] Now, we can factor the quadratic: \[ (t - 2)(t - 4) \leq 0 \] To find the intervals where this inequality holds, we can determine the roots of the equation \( (t - 2)(t - 4) = 0 \): \[ t = 2 \quad \text{and} \quad t = 4 \] Next, we can test the intervals defined by these roots: \( (-\infty, 2) \), \( (2, 4) \), and \( (4, \infty) \). 1. **For \( t < 2 \)** (e.g., \( t = 0 \)): \[ (0 - 2)(0 - 4) = 8 > 0 \] 2. **For \( 2 < t < 4 \)** (e.g., \( t = 3 \)): \[ (3 - 2)(3 - 4) = -1 < 0 \] 3. **For \( t > 4 \)** (e.g., \( t = 5 \)): \[ (5 - 2)(5 - 4) = 3 > 0 \] The inequality \( (t - 2)(t - 4) \leq 0 \) holds true in the interval \( [2, 4] \). Thus, the projectile is at least 128 feet above the ground during the time period: \[ \boxed{[2, 4]} \]