Final answer:
The height of the ball when it was thrown is 1.456 meters. The maximum height of the football is 9.1 meters. The ball hits the ground at either 0.416 seconds or 2.084 seconds.
Step-by-step explanation:
(a) To find the initial height of the ball, we can substitute t = 0 into the height equation h(t) = -4.9(t-1.25)² + 9.1. Thus, the height of the ball when it was thrown is h(0) = -4.9(0-1.25)² + 9.1 = -4.9(1.25)² + 9.1 = -4.9(1.56) + 9.1 = -7.644 + 9.1 = 1.456 meters.
(b) The maximum height of the football can be found by determining the vertex of the quadratic function. The vertex occurs at t = 1.25 seconds, and substituting this value into the height equation gives us h(1.25) = -4.9(1.25-1.25)² + 9.1 = -4.9(0)² + 9.1 = 0 + 9.1 = 9.1 meters.
(c) To find the height of the ball at t = 2.5 seconds, we substitute this value into the height equation: h(2.5) = -4.9(2.5-1.25)² + 9.1 = -4.9(1.25)² + 9.1 = -4.9(1.56) + 9.1 = -7.644 + 9.1 = 1.456 meters.
(d) To determine if the football is in the air after 6 seconds, we check if the height is positive at t = 6. Evaluating h(6) = -4.9(6-1.25)² + 9.1 = -4.9(4.75)² + 9.1 = -4.9(22.5625) + 9.1 = -110.3475 + 9.1 = -101.2475 meters. Since the height is negative, the football is not in the air after 6 seconds.
(e) The ball hits the ground when its height is zero. We can set the height equation h(t) = 0 and solve for t: 0 = -4.9(t-1.25)² + 9.1. This is a quadratic equation, and solving for t yields two solutions: t = 0.416 seconds and t = 2.084 seconds. The ball hits the ground at either 0.416 seconds or 2.084 seconds.