- The function \(y = 80x + 2\) models distance \(y\) in miles.
- Dimensional analysis shows that multiplying speed (miles per hour) by \(x\) must result in miles.
- The unit of \(x\) must therefore be hours, meaning the \(x\)-axis represents time.
- Final answer: \(\boxed{\text{time in hours}}\).
Explanation
- Problem Analysis
The problem describes a road trip using the linear function \(y = 80x + 2\). We are given that \(y\) represents the distance in miles from Carli's home, and the value \(2\) represents the starting distance from her home in miles. The coefficient \(80\) is associated with her speed, which is given in miles per hour.
- Identifying the Variable x
To determine what the variable \(x\) represents, we look at the units of the equation \(y = 80x + 2\). The equation represents a distance calculation where the total distance \(y\) (measured in miles) is the sum of the initial distance (also in miles) and the product of speed and another quantity. The term \(80\) has units of miles per hour (mph). For the product \(80 \cdot x\) to represent a distance in miles, the variable \(x\) must have units that, when multiplied by miles per hour, result in miles. Specifically, \([\text{miles}] = [\text{miles/hour}] \times [\text{hours}]\). Therefore, \(x\) must represent time measured in hours.
- Conclusion
Comparing our findings to the given options, we confirm that \(x\) represents time in hours. The other options, direction traveled and speed traveled, do not fit the dimensional analysis of the linear function model. Thus, the x-axis represents time in hours.
Examples
In physics and everyday navigation, linear equations are essential for predicting locations based on travel time. If you know you are driving at a constant speed of 50 miles per hour and start 10 miles from a landmark, the distance \(d\) from that landmark after \(t\) hours can be modeled as \(d = 50t + 10\). This allows you to plan your arrival times accurately and understand how travel time relates to distance covered.