- Identify the initial amount of water as the y-intercept \(b = 40\) and the rate of drainage as the negative slope \(m = -1.5\).
- Use the slope-intercept form \(y = mx + b\) to construct the linear function.
- Substitute the identified values to get the final equation: \(y = -1.5x + 40\).
- The final expression is \(\boxed{y = -1.5x + 40}\).
Explanation
- Problem Analysis
We are analyzing the relationship between the time \(x\) (in minutes) that a bathtub has been draining and the amount of water \(y\) (in gallons) remaining in it. The given data points are \((0, 40)\), \((0.5, 39.25)\), \((1, 38.5)\), and \((1.5, 37.75)\). We are also told the bathtub drains at a constant rate of \(1.5\) gallons per minute. Our goal is to derive the linear function \(y(x)\) representing the amount of water remaining at any time \(x\).
- Determining Parameters
A linear function can be written in the slope-intercept form \(y = mx + b\), where \(m\) represents the rate of change (the slope) and \(b\) is the initial value (the y-intercept). Given that the water is draining at a rate of \(1.5\) gallons per minute, the amount of water is decreasing over time. This means the rate of change is negative: \(m = -1.5\). The initial amount of water, which occurs when \(x = 0\), is \(40\) gallons. Thus, the y-intercept is \(b = 40\).
- Verification and Final Function
Substituting the values \(m = -1.5\) and \(b = 40\) into the slope-intercept form, we obtain the function:
\[y = -1.5x + 40\]
To ensure this function is correct, we can verify it with the provided data points.
For \(x = 0.5\): \(y = -1.5(0.5) + 40 = -0.75 + 40 = 39.25\).
For \(x = 1\): \(y = -1.5(1) + 40 = -1.5 + 40 = 38.5\).
For \(x = 1.5\): \(y = -1.5(1.5) + 40 = -2.25 + 40 = 37.75\).
All data points are consistent with the function.
Examples
Linear relationships like this are essential for planning and resource management. For instance, if you are filling a swimming pool or managing fuel consumption in a vehicle, you can model how the level of a liquid changes over time. By knowing the flow rate and the initial amount, you can accurately predict how much resource will remain at any given moment, helping you avoid running out or determine when you need to refill.