Certainly! Let's understand and solve the problem step-by-step.
The height of the soccer ball as a function of time [tex]\( t \)[/tex] is given by:
[tex]\[ h(t) = -16t^2 + 32t \][/tex]
To determine when the soccer ball is moving through the air, we need to find the time interval during which the height [tex]\( h(t) \)[/tex] is greater than zero.
### Step 1: Find when the ball touches the ground
The ball touches the ground when [tex]\( h(t) = 0 \)[/tex]. So solve for [tex]\( t \)[/tex] such that:
[tex]\[ -16t^2 + 32t = 0 \][/tex]
Factorizing the equation, we get:
[tex]\[ -16t(t - 2) = 0 \][/tex]
This gives us two solutions:
[tex]\[ t = 0 \quad \text{or} \quad t = 2 \][/tex]
### Step 2: Determine the time interval when the ball is in the air
From the solutions above, [tex]\( t = 0 \)[/tex] corresponds to the initial time when the ball is kicked from the ground, and [tex]\( t = 2 \)[/tex] corresponds to the time when the ball returns to the ground.
Therefore, the soccer ball is in the air from [tex]\( t = 0 \)[/tex] to [tex]\( t = 2 \)[/tex], but not including these points since at those times, the height is zero.
### Conclusion
The correct interval during which the soccer ball is moving through the air is:
[tex]\[ \{ t \mid 0 < t < 2 \} \][/tex]
So, the correct answer is:
[tex]\[ \{t \mid 0 < t < 2\} \][/tex]
