Given
The height in feet of a rock thrown from a bridge is s(t) = -16t² +32t +75.
The height of the bridge is 75 ft.
Find
(a) In interval notation, the time(s) the rock is lower than the bridge.
(b) An explanation why (-∞, 0) is not included in the solution.
Solution
Since the height of the bridge is 75 ft, the rock will be lower than that when ...
... s(t) < 75
... -16t² + 32t + 75 < 75 . . . . . substituting for s(t)
... -16t(t -2) < 0 . . . . . . . . . . . . subtract 75 and factor
This product changes sign at t=0 and t=2. It is negative for t < 0 and t > 2. The period t < 0 is not part of the reasonable domain of the function s(t), so is not included in the solution.
(a) The rock is lower than the bridge for t in the interval (2, ∞).
(b) The position of the rock is only reasonably defined by s(t) for t ≥ 0 (and less than about 3.4). Thus, the "solution" t ∈ (-∞, 0) is not reasonably included in the solution set.
