Find all numbers that satisfy the following.
-An arrow is fired straight up from the ground with an initial velocity of 96 feet per second. Its height, $s ( t )$, in feet at any time $t$ is given by the function $s ( t ) = - 16 t ^ { 2 } + 96 t$. Find the interval of time for which the height of the arrow is less than 80 feet.
A) after $5 \mathrm { sec }$
B) between 1 and $5 \mathrm { sec }$
C) before $1 \mathrm { sec }$
D) before $1 \mathrm { sec }$ or after $5 \mathrm { sec }$

Question

Quizplus · Accepted Answer

To find when the height of the arrow is less than 80 feet, we set the height function less than 80 and solve for $t$: $-16t^2 + 96t < 80$. Rearranging gives $-16t^2 + 96t - 80 < 0$, which simplifies to $16t^2 - 96t + 80 > 0$ (by multiplying both sides by -1 and flipping the inequality). Factoring out 16 gives $t^2 - 6t + 5 > 0$, which factors further to $(t-1)(t-5) > 0$. This inequality is satisfied before $t = 1$ and after $t = 5$, indicating the arrow is below 80 feet before 1 second and after 5 seconds.

Find All Numbers That Satisfy the Following s(t)s ( t )s(t)

Find All Numbers That Satisfy the Following $s ( t )$