Question

question 11 (4 points) a ball is thrown vertically upward with an initial velocity of 96 feet per second. the distance s (in feet) of the ball from the ground after t seconds is ( s(t) = 96t - 16t^2 ). for what time(s) t is the ball more than 128 feet above the ground? you must show all of your work to algebraically solve the problem by uploading your handwritten work. you cannot simply graph and solve.

Explanation:

Step1: Set up inequality for height

$96t - 16t^2 > 128$

Step2: Rearrange to standard quadratic form

$-16t^2 + 96t - 128 > 0$
Divide by $-16$ (reverse inequality):
$t^2 - 6t + 8 < 0$

Step3: Factor the quadratic

$(t-2)(t-4) < 0$

Step4: Find critical points and test intervals

Critical points: $t=2$, $t=4$
Test intervals:

• $t<2$: $(negative)(negative)=positive$ (not <0)
• $2
• $t>4$: $(positive)(positive)=positive$ (not <0)

Answer:

The ball is more than 128 feet above the ground when $2 < t < 4$ seconds.