To determine the amount of time it takes for the nail to reach the same orientation it had when it entered the tire, we need to identify the key feature of the function that describes the nail's travel with respect to time. This key feature is the period of the function, which indicates the duration required for a repeating event or cycle.
Let's go through the data step by step to understand how we can find this period:
1. Understand the Table: The table provides the time in seconds and the corresponding height of the nail off the ground in inches. The heights are measured at specific intervals, showing the nail’s vertical position at different instances of time.
2. Identify the Cycle: A period is the time interval in which a repeating cycle of a waveform or function completes. Since the height of the nail is cyclic as it moves with the rotation of the tire, the table should reflect a repeating sequence of heights.
3. Analyze the Heights: By observing the heights in the table, we can see the pattern of the nail’s travel. The heights start from 0 inches, increase to a maximum value (25.5 inches) and then decrease back to the initial height (21.5 inches) within the given times.
4. Determine the Starting and Ending Points of the Cycle: The table starts at time [tex]\(0\)[/tex] seconds. To find when the nail reaches the same orientation, we need to identify when the height pattern begins to repeat. In this case, the data shows a return towards 21.5 inches, which indicates it's near the end of a cycle.
- At time [tex]\(0.07\)[/tex] seconds, the height is 21.5 inches, which suggests completion of nearly one full cycle.
5. Calculate the Period: The period of this function can be found by subtracting the starting time from the ending time of the analyzed cycle:
[tex]\[
\text{Period} = \text{End Time} - \text{Start Time} = 0.07 \ \text{seconds}
\][/tex]
Thus, the period of the function representing the nail's travel is 0.07 seconds. This is the amount of time it takes for the nail to complete one full cycle and return to the same orientation it had when it entered the tire.
