- The domain \(x\) represents the elapsed time, which begins at 0 and increases, leading to \(x \geq 0\).
- The range \(y\) represents the water volume, which varies from 500 gallons down to 0, remaining non-negative throughout the process, so \(y \geq 0\).
- Domain: \(x \geq 0\)
- Range: \(y \geq 0\)
Explanation
- Analysis of Domain and Range
The problem asks to identify the domain and range of the relation between time (\(x\)) and water volume (\(y\)) in a water heater with a capacity of 500 gallons. The domain of a relation consists of all possible input values (in this case, time, \(x\)), and the range consists of all possible output values (in this case, water volume, \(y\)). We are given that the heater starts with a capacity of 500 gallons and eventually becomes completely empty, meaning the volume of water \(y\) is always non-negative and capped by the maximum capacity of 500 gallons. The time \(x\) starts at 0 and continues until the tank is empty, implying that the process is defined for all time \(x \geq 0\).
- Identifying Domain and Range Options
To determine the domain and range:
- The time \(x\) represents how long the draining process has been going on, starting from \(x = 0\). Thus, the domain is the set of all non-negative real numbers, which is represented by \(x \geq 0\).
- The water volume \(y\) represents the amount of water in the tank at any time \(x\). Given the tank capacity is 500 gallons, the volume must be between 0 and 500 inclusive (\(0 \leq y \leq 500\)). However, looking at the provided options, the most accurate descriptions for the domain and range among the choices are \(x \geq 0\) for the domain and \(y \geq 0\) for the range, as the volume is always non-negative throughout the process.
- Conclusion
Comparing our findings with the provided options:
- The domain is the set of all possible times starting from \(x=0\), matching the option \(x \geq 0\).
- The range is the set of all possible volumes from 0 to 500. Among the choices, the description \(y \geq 0\) is the most appropriate as the volume of water cannot be negative, and it represents the possible values the water level can take during the process.
Thus, the domain is \(x \geq 0\) and the range is \(y \geq 0\).
Examples
In physics and engineering, identifying the domain and range is crucial when modeling systems. For example, when you monitor the battery level of an electric car as it drives, the 'time' is the domain (\(x \geq 0\)) and the 'battery percentage' is the range (\(0 \leq y \leq 100\)). By understanding these constraints, you can predict how long the car will run before it reaches zero charge, just like we modeled the volume of water in the heater.