- The height is given by \(A(t) = -16t^2 + 32t\).
- To find when the ball is in the air, solve the inequality \(-16t^2 + 32t > 0\).
- Factoring gives \(-16t(t-2) > 0\), which simplifies to \(t(t-2) < 0\).
- The solution is the interval \(\{t \mid 0 < t < 2\}\), which is \(\boxed{\{t \mid 0 < t < 2\}}\).
Explanation
- Problem Analysis
The height of the soccer ball at time \(t\) is modeled by the function \(A(t) = -16t^2 + 32t\), where \(t\) represents the time in seconds and \(A(t)\) represents the height in feet. For the ball to be "moving through the air," its height above the ground must be strictly positive, meaning \(A(t) > 0\). We are asked to find the time interval during which this condition is satisfied.
- Setting Up and Solving the Inequality
To find the interval, we set up the inequality:
\[-16t^2 + 32t > 0\]
First, we factor out the common term \(-16t\):
\[-16t(t - 2) > 0\]
Alternatively, we can divide the inequality by \(-16\), which reverses the direction of the inequality sign:
\[t^2 - 2t < 0\]
Factoring this gives:
\[t(t - 2) < 0\]
- Determining the Interval
The roots of the quadratic equation \(t(t - 2) = 0\) are \(t = 0\) and \(t = 2\). These points divide the number line into three intervals: \(t < 0\), \(0 < t < 2\), and \(t > 2\). Since we want the product \(t(t-2)\) to be negative, we test values in each interval. For \(0 < t < 2\), we can pick \(t = 1\), which gives \(1(1-2) = -1 < 0\). Thus, the interval where the height is positive is \(0 < t < 2\). Since the ball is kicked at \(t=0\), this represents the duration of the ball's flight.
- Conclusion
Comparing our result \(0 < t < 2\) with the given options, the correct interval is \(\{t \mid 0 < t < 2\}\). Thus, the soccer ball is moving through the air for this time interval. The final answer is \(\boxed{\{t \mid 0 < t < 2\}}\).
Examples
This model of a projectile's path is a classic application of quadratic functions. In real life, architects or engineers use similar equations to calculate the path of water in a fountain or the trajectory of an object launched from a specific point. Understanding these intervals helps predict when an object will hit the ground or reach its peak height, which is essential for design and safety.