Arock is thrown straight up from the top of a bridge that is 75 ft high with an initial velocity of 32 ft/s. the height of the object can be modeled by the equation s(t) = -16t2 + 32t + 75. determine the time(s) the ball is lower than the bridge. write your answer in interval notation. in two or more complete sentences, explain why is(-∞,0) not included in the solution.

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.

choice b is correct

step-by-step explanation:

we have been given the following system of inequalities;

we are required to graph the solution to the system of inequalities. using a graphing tool, the graph of the system is as shown in the attachment. the continuous line represents the graph of while the dotted line represents .