To determine how long it will take for the object to hit the ground, we'll use the given height function:
[tex]\[
h(t) = -16t^2 + 144
\][/tex]
Here, [tex]\( h(t) \)[/tex] represents the height of the object at time [tex]\( t \)[/tex]. We need to find the time [tex]\( t \)[/tex] when the object hits the ground, which means [tex]\( h(t) = 0 \)[/tex].
Let's set the equation to zero and solve for [tex]\( t \)[/tex]:
[tex]\[
-16t^2 + 144 = 0
\][/tex]
First, move 144 to the other side of the equation:
[tex]\[
-16t^2 = -144
\][/tex]
Next, divide both sides of the equation by -16 to isolate [tex]\( t^2 \)[/tex]:
[tex]\[
t^2 = \frac{144}{16}
\][/tex]
Simplify the fraction:
[tex]\[
t^2 = 9
\][/tex]
To find [tex]\( t \)[/tex], take the square root of both sides:
[tex]\[
t = \sqrt{9}
\][/tex]
Thus, we have two potential values for [tex]\( t \)[/tex]:
[tex]\[
t = 3 \quad \text{or} \quad t = -3
\][/tex]
Since time cannot be negative in this context, we discard [tex]\( t = -3 \)[/tex]. Thus, the time [tex]\( t \)[/tex] when the object hits the ground is:
[tex]\[
t = 3 \text{ seconds}
\][/tex]
So, it will take 3 seconds for the object to hit the ground.
