- The water requirement is modeled by the inequality \(y \geq 0.8x + 1.5\).
- To find a valid solution \((x, y)\), choose a duration \(x \geq 0\) and ensure \(y \geq 0.8x + 1.5\).
- For a 2-hour hike (\(x=2\)), the minimum water required is \(0.8(2) + 1.5 = 3.1\) liters.
- Any \((x, y)\) satisfying the given inequality is a solution, represented by \(\boxed{y \geq 0.8x + 1.5}\).
Explanation
- Understanding the Problem and Data Analysis
We are given that the amount of water \(y\) (in liters) required for a hike of duration \(x\) (in hours) is modeled by the linear inequality \(y \geq 0.8x + 1.5\). Here, \(x\) and \(y\) must be non-negative real numbers since duration and volume cannot be negative. This inequality tells us that the total water required is at least the sum of the consumption rate (0.8 liters per hour) and the safety reserve (1.5 liters). To find a valid solution, we need to choose a duration \(x \geq 0\) and then ensure that the amount of water \(y\) is greater than or equal to the minimum requirement \(0.8x + 1.5\).
- Determining and Verifying Solutions
To determine if a pair \((x, y)\) is a solution, we substitute the value of \(x\) into the expression \(0.8x + 1.5\) to find the minimum required water volume, \(y_{min}\). For example, if we consider a hike of duration \(x = 2\) hours:
- Calculate the minimum water required: \(y_{min} = 0.8(2) + 1.5 = 1.6 + 1.5 = 3.1\) liters.
- Any total water amount \(y \geq 3.1\) liters will satisfy the requirement.
For instance, if Erin takes \(y = 3.5\) liters of water for a \(x = 2\) hour hike, the condition \(3.5 \geq 0.8(2) + 1.5\) becomes \(3.5 \geq 3.1\), which is true. Thus, \((2, 3.5)\) is a valid solution.
- Conclusion and Final Answer
In conclusion, any pair of non-negative real numbers \((x, y)\) such that \(y \geq 0.8x + 1.5\) is a valid solution. One such valid solution is the pair \((2, 3.5)\), meaning for a 2-hour hike, carrying 3.5 liters of water is sufficient to meet the requirement. Generally, the set of all solutions is represented by the region on or above the line \(y = 0.8x + 1.5\) in the first quadrant of the coordinate plane. The final answer is the inequality itself, as it defines the set of all valid solutions: \(\boxed{y \geq 0.8x + 1.5}\).
Examples
Imagine you are planning a road trip and you need to ensure you have enough fuel. If your car consumes 0.05 gallons per mile and you want to keep at least 2 gallons in the tank as a reserve, you can model your fuel requirement with the inequality \(f \geq 0.05d + 2\), where \(f\) is the total gallons needed and \(d\) is the distance in miles. By using this model, you can determine exactly how much fuel to put in your tank to stay safe during your journey, just as Erin does with her water supply.