Find all numbers that satisfy the following.
-An arrow is fired straight up from the ground with an initial velocity of 128 feet per second. Its height, $\mathrm { s } ( \mathrm { t } )$, in feet at any time $\mathrm { t }$ is given by the function $\mathrm { s } ( \mathrm { t } ) = - 16 \mathrm { t } ^ { 2 } + 128 \mathrm { t }$. Find the interval of time for which the height of the arrow is greater than 156 feet.
A) after $\frac { 3 } { 2 } \mathrm { sec }$
B) between $\frac { 3 } { 2 }$ and $\frac { 13 } { 2 } \mathrm { sec }$
C) before $\frac { 3 } { 2 }$ sec or after $\frac { 13 } { 2 }$ sec
D) before $\frac { 13 } { 2 } \mathrm { sec }$

Question

Quizplus · Accepted Answer

To find the interval of time for which the height of the arrow is greater than 156 feet, we set the height function greater than 156 and solve for $t$: $-16t^2 + 128t > 156$. Rearranging gives $-16t^2 + 128t - 156 > 0$, which simplifies to $-16(t^2 - 8t + 9.75) > 0$. Factoring or using the quadratic formula, we find the roots of the equation $t^2 - 8t + 9.75 = 0$ to be $t = \frac{3}{2}$ and $t = \frac{13}{2}$. Since the parabola opens downwards (the coefficient of $t^2$ is negative), the height is greater than 156 feet between these two times.

Find All Numbers That Satisfy the Following s(t)\mathrm { s } ( \mathrm { t } )s(t)

Find All Numbers That Satisfy the Following $\mathrm { s } ( \mathrm { t } )$