Final answer:
The ball's height above the ground is represented by the quadratic equation s(t) = -16t² + 126t, and to find the times when it is 227 feet high, we use the quadratic formula with the coefficients from this equation. After solving, we discard negative values and conclude that there are usually two times that the ball is at the same height—once ascending and once descending.
Step-by-step explanation:
The student is asking for the times at which a ball will be 227 feet from the ground after being projected upward. The ball's height above the ground is given by the quadratic equation s(t) = -16t² + 126t. To find when the ball is 227 feet high, we set the equation equal to 227 and solve for t:
227 = -16t² + 126t.
This is a quadratic equation that we can solve using the quadratic formula, where a = -16, b = 126, and c = -227. The quadratic formula is given by:
t = −b ± √(b² - 4ac) / (2a)
Substituting the values, we find two possible times for when the ball is at 227 feet from the ground. We discard any negative time values since they don't make physical sense in this context. Thus, we are left with two possible times, which typically represent the ball being at the same height on the way up and on the way down.