To determine the time it will take for the camera to hit the ground, we need to solve for [tex]\( t \)[/tex] when the height [tex]\( h(t) \)[/tex] is 0. The equation for height as a function of time is given by:
[tex]\[
h(t) = -16t^2 + \text{initial height}
\][/tex]
Given that the initial height is [tex]\( 4096 \)[/tex] feet, the equation becomes:
[tex]\[
h(t) = -16t^2 + 4096
\][/tex]
We set [tex]\( h(t) \)[/tex] to 0 since we're looking for when the camera hits the ground:
[tex]\[
0 = -16t^2 + 4096
\][/tex]
Next, we solve for [tex]\( t \)[/tex]. Start by isolating the term involving [tex]\( t \)[/tex]:
[tex]\[
-16t^2 + 4096 = 0
\][/tex]
Subtract 4096 from both sides:
[tex]\[
-16t^2 = -4096
\][/tex]
Next, divide both sides by -16 to solve for [tex]\( t^2 \)[/tex]:
[tex]\[
t^2 = \frac{4096}{16}
\][/tex]
Calculate the right-hand side:
[tex]\[
t^2 = 256
\][/tex]
To find [tex]\( t \)[/tex], take the square root of both sides:
[tex]\[
t = \sqrt{256}
\][/tex]
[tex]\[
t = 16
\][/tex]
Therefore, it will take [tex]\( 16 \)[/tex] seconds for the camera to hit the ground.
[tex]\(\boxed{16}\)[/tex] seconds
