To solve for when the projectile hits the target, we need to find the value of t at which the height of the projectile is equal to 8 feet. We can set the height function equal to 8 and solve for t:
d(t) = -1.2t^2 + 9t + 2.5 (where d is the height of the projectile at time t)
8 = -1.2t^2 + 9t + 2.5 (substitute 8 for d(t))
0 = -1.2t^2 + 9t + 2.5 - 8 (rearrange and simplify)
0 = -1.2t^2 + 9t - 5.5
We can solve for t using the quadratic formula:
t = (-b ± sqrt(b^2 - 4ac)) / 2a
where a = -1.2, b = 9, and c = -5.5
t = (-9 ± sqrt(9^2 - 4(-1.2)(-5.5))) / 2(-1.2)
t = (-9 ± sqrt(121.6)) / -2.4
t = (-9 ± 11) / -2.4
The two solutions for t are:
t = 0.833 seconds and t = 6.667 seconds
Since the projectile is on its way back down when it hits the target, we can discard the larger solution (6.667 seconds) and conclude that the projectile will hit the target after 0.833 seconds.