The height (in meters) of a shot cannonball follows a trajectory given by $h(t) = -4.9t^2 + 14t - 0.4$ at time $t$ (in seconds). As an improper fraction, for how long is the cannonball above a height of $6$ meters?
The cannonball is above a height of 6 meters when h(t)≥6. Solving this inequality, we get −4.9t2+14t−5.6≥0. Factoring the left-hand side, we get (−4.9t+7)(t−0.8)≥0. This inequality is satisfied when −4.97≤t≤0.8. Converting these inequalities to improper fractions, we get 4930≤t≤98. Therefore, the cannonball is above a height of 6 meters for a total time of 8/9 − 30/49 = 151/441 seconds.
