- Identify the initial state of the bathtub, which is \(y = 40\) at \(x = 0\).
- Model the drainage as a linear function \(y = 40 - 1.5x\), where \(x\) represents time in minutes.
- Determine the minimum value of \(y\) when the tub is empty, which is \(y = 0\).
- Conclude that the range of the function is all real numbers such that \(\boxed{0 \leq y \leq 40}\).
Explanation
- Establishing the Linear Relationship
We are given a scenario where a bathtub is draining water at a constant rate. To find the range of the function representing the amount of water \(y\) over time \(x\), we first identify the linear relationship between the volume of water and time. The initial amount of water in the bathtub at \(x = 0\) minutes is given as \(y = 40\) gallons. The draining rate is 1.5 gallons per minute. Thus, the relationship can be expressed by the linear function:
\[y(x) = 40 - 1.5x\]
The domain of this function, in a physical sense, is restricted to \(x \geq 0\), meaning time cannot be negative. Additionally, the process of draining stops when the bathtub is empty, which corresponds to \(y = 0\).
- Determining the Range of the Function
To determine the range of the function, we find the values of \(y\) that correspond to the valid domain of \(x\). The maximum value of \(y\) occurs at the start, when \(x = 0\):
\[y(0) = 40 - 1.5(0) = 40\]
The minimum value of \(y\) occurs when the bathtub is empty, which is when \(y = 0\). We find the time at which this happens by solving the equation:
\[40 - 1.5x = 0\]
\[1.5x = 40\]
\[x = \frac{40}{1.5} = \frac{80}{3} \approx 26.67 \text{ minutes}\]
Since the function is linear and decreasing, the amount of water \(y\) continuously decreases from 40 gallons to 0 gallons as the time \(x\) increases from 0 to approximately 26.67 minutes. Thus, the possible values for \(y\) cover all real numbers from 0 to 40 inclusive.
- Conclusion
Comparing our finding to the provided options, we can see that the range is defined by the interval \([0, 40]\). Therefore, the correct range is all real numbers such that \(0 \leq y \leq 40\). The final answer is \(\boxed{0 \leq y \leq 40}\).
Examples
Understanding ranges of functions is essential in daily life for resource management. For example, if you are filling a gas tank with a capacity of 15 gallons, the amount of fuel \(y\) in your tank as you fill it at a rate of 3 gallons per minute can be modeled by a linear function \(y = 3t\). Knowing that your tank starts empty (\(y=0\)) and fills to 15 gallons (\(y=15\)), you can determine that the range of the fuel level function is \(0 \leq y \leq 15\). This helps you calculate how much time it takes to fill the tank or how much fuel you currently have based on the time you've spent at the pump.