To solve these problems, we'll start with the height function given by [tex]\( h(t) = 96t - 16t^2 \)[/tex], where [tex]\( h(t) \)[/tex] represents the height in feet and [tex]\( t \)[/tex] represents the time in seconds.
### Finding When the Object is 144 Feet in the Air
We need to find the time [tex]\( t \)[/tex] when the height [tex]\( h(t) \)[/tex] is 144 feet. Setting [tex]\( h(t) = 144 \)[/tex], we get the equation:
[tex]\[
96t - 16t^2 = 144
\][/tex]
Rearranging this equation, we bring all terms to one side:
[tex]\[
16t^2 - 96t + 144 = 0
\][/tex]
Next, we divide the entire equation by 16 to simplify:
[tex]\[
t^2 - 6t + 9 = 0
\][/tex]
This simplifies to a quadratic equation that can be solved by factoring:
[tex]\[
(t - 3)^2 = 0
\][/tex]
Solving for [tex]\( t \)[/tex], we get:
[tex]\[
t = 3
\][/tex]
Therefore, the object will be 144 feet in the air at [tex]\( t = 3 \)[/tex] seconds.
### Finding When the Object Hits the Ground
To determine when the object hits the ground, we set [tex]\( h(t) = 0 \)[/tex]:
[tex]\[
96t - 16t^2 = 0
\][/tex]
We can factor out the common term [tex]\( t \)[/tex]:
[tex]\[
t(96 - 16t) = 0
\][/tex]
Setting each factor to zero gives us:
[tex]\[
t = 0 \quad \text{or} \quad 96 - 16t = 0
\][/tex]
Solving the second equation for [tex]\( t \)[/tex]:
[tex]\[
96 = 16t \\
t = \frac{96}{16} \\
t = 6
\][/tex]
Therefore, the object hits the ground at [tex]\( t = 6 \)[/tex] seconds.
### Summary
- The object will be 144 feet in the air at [tex]\( t = 3 \)[/tex] seconds.
- The object will hit the ground at [tex]\( t = 0 \)[/tex] seconds (launch time) and [tex]\( t = 6 \)[/tex] seconds.
