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.

maximum area = xy = (16)(32) = 512 ft2

to maximize the area of the garden, the side of the fence parallel to the side of the house should be 32 ft long, and the other two sides should both have length 16 ft.'

source

no because after the troops were pulled many blacks were killed by hangings lynchings and other cruel ways which in a way interrupted the reconstruction of the south and differed from the way abraham lincoln had an idea of reconstruction