- Analyze the given height data: the height increases from \(t=0\) to \(t=2\) and decreases from \(t=2\) to \(t=4\).
- Compare the trends against the definitions of models: exponential models exhibit a constant ratio, while quadratic models exhibit a parabolic shape (rising and falling).
- Conclude that the "increase and then decrease" behavior identifies the model as quadratic.
- Final answer: \(\boxed{\text{quadratic, because the height increases and then decreases}}\)
Explanation
- Analyzing the Data Trends
We analyze the given data to determine which mathematical model (quadratic or exponential) best fits the object's height over time. The data provided is:
- At \(t = 0\), \(h = 5\)
- At \(t = 1\), \(h = 50\)
- At \(t = 2\), \(h = 70\)
- At \(t = 3\), \(h = 48\)
- At \(t = 4\), \(h = 4\)
By observing the trend, we see that the height increases from \(5\) to \(70\) as time progresses from \(0\) to \(2\) seconds, and then it decreases from \(70\) to \(4\) as time progresses from \(2\) to \(4\) seconds. This "increase and then decrease" behavior is characteristic of a parabolic arc, which is a key feature of quadratic functions. In contrast, standard exponential models typically demonstrate monotonic growth or decay and do not return to near their initial values after reaching a peak.
- Evaluating the Model Characteristics
To distinguish between the two models, we check the properties of quadratic and exponential growth:
- Exponential models: Defined by a constant multiplicative rate of change (e.g., multiplying the previous value by a constant factor). Looking at the ratios between successive heights: \(\frac{50}{5} = 10\), \(\frac{70}{50} = 1.4\), \(\frac{48}{70} \approx 0.69\). Since the ratio is not constant, the data does not follow a simple exponential model.
- Quadratic models: Typically represented by a second-order polynomial \(h(t) = at^2 + bt + c\). A defining feature is that the graph forms a parabola, which naturally rises to a maximum and then falls. Given the data's clear peak at \(t = 2\) and its symmetric-like increase and decrease, it strongly aligns with a quadratic trajectory. Thus, the model that best represents this data is quadratic, specifically because the object's height increases to a peak and then decreases.
- Final Conclusion
Comparing our findings with the provided options, we conclude that the data represents a quadratic model because the height increases and then decreases. While multiplicative rates of change are characteristic of exponential models, the observed behavior of the height (rising then falling) is the defining attribute of the quadratic model for this scenario. Therefore, the second option is the correct classification.
Examples
This scenario is a classic example of projectile motion. In physics, when an object is thrown into the air, its height follows a parabolic path described by the equation \(h(t) = -gt^2 + v_0t + h_0\). By observing that the height increases and then decreases, we can use quadratic modeling to predict when the object will hit the ground or reach its maximum height, which is essential for engineering, sports analysis, and safety planning.