A GPS device tracks the elevation $E$ (in feet) of a hiker walking in the mountains. The elevation $t$ hours after beginning the hike is given in the figure. <IMAGE>
Notice that the curve in the figure is horizontal for an interval of time near $t=5.5$ hr. Give a plausible explanation for the horizontal line segment.

Verified step by step guidance

1. Identify that a horizontal line segment on a graph of elevation versus time indicates that the elevation is constant over that interval.
2. Understand that a constant elevation means there is no change in height, implying the hiker is neither ascending nor descending.
3. Consider possible real-world scenarios: the hiker could be taking a break, resting, or walking on a flat section of the trail.
4. Recognize that during this time, the derivative of the elevation with respect to time, which represents the rate of change of elevation, is zero.
5. Conclude that the horizontal segment suggests a period of no elevation change, possibly due to a rest or a flat terrain, which is common in hiking scenarios.

Key Concepts

Derivative and Rate of Change

The derivative of a function represents the rate of change of that function with respect to its variable. In the context of the elevation function E(t), the derivative E'(t) indicates how the elevation changes as time progresses. A horizontal line segment on the graph implies that the derivative is zero over that interval, meaning the elevation remains constant during that time.

Horizontal Line Segment

A horizontal line segment on a graph indicates that the dependent variable (in this case, elevation) does not change as the independent variable (time) changes. This suggests that the hiker is at a constant elevation, which could occur when they are resting, walking on flat terrain, or navigating a plateau. Understanding this concept helps explain the behavior of the hiker's elevation over time.

Interpretation of Graphs

Interpreting graphs involves analyzing the visual representation of data to understand relationships between variables. In this scenario, the graph of elevation versus time allows us to observe how the hiker's elevation changes. Recognizing features like horizontal segments is crucial for making inferences about the hiker's activity, such as periods of rest or flat walking.