- The problem asks for the interval of time \(t\) for which the car is in the air, defined by \(d = 144 - 16t^2 > 0\).
- Solving the inequality \(144 - 16t^2 > 0\) leads to \(16t^2 < 144\), which simplifies to \(t^2 < 9\), implying \(t < 3\).
- Since time \(t \ge 0\), the car remains in the air during the interval \(0 \le t < 3\).
- The most suitable given interval is \(\boxed{0 < t < 3}\).
Explanation
- Understanding the Problem and Data Analysis
We are given that the distance \(d\) of a roller-coaster car from the ground as it descends is modeled by the equation \(d = 144 - 16t^2\), where \(t\) represents the time in seconds. We are looking for the interval of time during which the car is in the air. The car is "in the air" when its distance from the ground is greater than zero, so we must solve the inequality \(d > 0\).
- Solving the Inequality for Time \(t\)
To find the time interval during which the car is in the air, we set the distance equation to be greater than zero:
\[144 - 16t^2 > 0\]
Next, we solve this inequality for \(t\). First, we subtract \(144\) from both sides:
\[-16t^2 > -144\]
Dividing both sides by \(-16\) (and remembering to reverse the inequality sign when dividing by a negative number):
\[t^2 < \frac{-144}{-16}\]
\[t^2 < 9\]
Taking the square root of both sides, we find that the magnitude of \(t\) must be less than \(3\):
\[|t| < 3\]
Since \(t\) represents time elapsed during the descent, we consider \(t \geq 0\). Therefore, the inequality simplifies to:
\[0 \le t < 3\]
This means the car is above the ground from the start of the descent (\(t=0\)) until just before it reaches the ground (\(t=3\)). Among the given options, the interval \(0 < t < 3\) best represents the period during which the car is actively moving in the air before hitting the ground.
- Conclusion
Comparing our derived interval \(0 \le t < 3\) to the provided options, the most appropriate choice is \(0 < t < 3\). The car is at the top of the ride at \(t = 0\) and reaches the ground at \(t = 3\). Thus, the interval during which the car is moving in the air is \(\boxed{0 < t < 3}\).
Examples
Calculating the time a car spends in the air during a descent is a common application of kinematics in physics. Engineers use similar equations to model the motion of roller coasters, thrill rides, and even falling objects to ensure passenger safety. By determining exactly when a vehicle will reach the ground, designers can synchronize safety mechanisms and optimize the design for a smooth ride experience.