To determine which statement correctly describes when the drone is on the ground, we need to find the values of [tex]\( h(t) \)[/tex] at specific times [tex]\( t \)[/tex]. Let's analyze the height function given by:
[tex]\[
h(t) = -16(t-2)^2 + 64
\][/tex]
1. First, we calculate the height of the drone at [tex]\( t = 0 \)[/tex]:
[tex]\[
h(0) = -16(0-2)^2 + 64
\][/tex]
[tex]\[
h(0) = -16(4) + 64
\][/tex]
[tex]\[
h(0) = -64 + 64 = 0
\][/tex]
So, at [tex]\( t = 0 \)[/tex] seconds, the drone is on the ground ([tex]\( h(0) = 0 \)[/tex]).
2. Next, we calculate the height at [tex]\( t = 2 \)[/tex]:
[tex]\[
h(2) = -16(2-2)^2 + 64
\][/tex]
[tex]\[
h(2) = -16(0) + 64
\][/tex]
[tex]\[
h(2) = 64
\][/tex]
At [tex]\( t = 2 \)[/tex] seconds, the drone is at its maximum height ([tex]\( h(2) = 64 \)[/tex]).
3. Finally, we calculate the height at [tex]\( t = 4 \)[/tex]:
[tex]\[
h(4) = -16(4-2)^2 + 64
\][/tex]
[tex]\[
h(4) = -16(2)^2 + 64
\][/tex]
[tex]\[
h(4) = -64 + 64 = 0
\][/tex]
So, at [tex]\( t = 4 \)[/tex] seconds, the drone is on the ground again ([tex]\( h(4) = 0 \)[/tex]).
From these calculations, we see that the drone is on the ground at [tex]\( t = 0 \)[/tex] and [tex]\( t = 4 \)[/tex] seconds. This matches the statement in option D:
D. The drone is on the ground at 0 and 4 seconds.
Therefore, the correct answer is:
D. The drone is on the ground at 0 and 4 seconds.
