Problem 2 - Algebra 2

Question

A pump removes 1000 gal of water from a pool at a constant rate of 50 gal/min. a. Write an equation to find the amount of water \(y\) in the pool after \(t\) minutes. b. Graph the equation and interpret the \(t\) - and \(y\) -intercepts.

Step-by-Step Solution

Answer

The equation to find the amount of water y in gallons in the pool after t minutes is \( y = 1000 - 50t \). The pool will be empty after 20 minutes (t-intercept), and it initially contains 1000 gallons of water (y-intercept).

Step 1: Write the Equation

To write the equation, understand that the amount of water removed from the pool is directly proportional to the time spent pumping. In 1 minute, 50 gallons are removed. We can write the amount of water remaining after t minutes as follows: initially, there are 1000 gallons, and 50 gallons per minute are being pumped out, so after t minutes, the amount of water removed is 50t. Hence, the amount of water remaining, y, is 1000 minus the amount removed (50t). The equation for the amount of water in the pool after t minutes is: \( y = 1000 - 50t \).

Step 2: Interpret the t-intercept

The t-intercept of the equation is the value of t when the amount of water remaining, y, is 0. Setting the equation to 0 to find the t-intercept gives \( 1000 - 50t = 0 \). Solving for t, we get \( t = \frac{1000}{50} = 20 \) minutes. Thus, the t-intercept is (20, 0), indicating that in 20 minutes the pool will have no water left.

Step 3: Interpret the y-intercept

The y-intercept of the equation is the value of y when t is 0. In the equation \( y = 1000 - 50t \), if t is 0, then \( y = 1000 - 50(0) = 1000 \) gallons. Thus, the y-intercept is (0, 1000), representing the initial volume of water in the pool before any pumping.

Step 4: Graph the Equation

To graph the equation \( y = 1000 - 50t \), plot the t-intercept (20, 0) and the y-intercept (0, 1000) on a graph with t on the horizontal axis and y on the vertical axis. Draw a straight line connecting these points, which will slope downwards from left to right, indicating that as time passes, the amount of water in the pool decreases.

Key Concepts

Linear Equations

Understanding linear equations is fundamental in algebra, as they describe a relationship between two variables with a constant rate of change. A linear equation can be written in the form y = mx + b, where m represents the slope of the line, and b represents the y-intercept—the point where the line crosses the y-axis.

For the exercise given, we are looking at a swimming pool where the water is being pumped out at a constant rate. The linear equation to represent this scenario is y = 1000 - 50t. Here, y is the amount of water in the pool after t minutes, 1000 is the initial amount of water, and 50t represents the volume of water that has been removed after t minutes.

Graphing Linear Equations

Graphing a linear equation visually represents the relationship between two variables. To graph the equation \( y = 1000 - 50t \), you start by plotting the y-intercept and the t-intercept on their respective axes.

For the graphing process, it's important to note two crucial points that will define the line: the y-intercept (0, 1000) and the t-intercept (20, 0). After plotting these on a Cartesian plane, draw a line through these two points. The line you draw is the graph of the equation, and it showcases how the amount of water in the pool decreases over time.

Y-Intercept

The y-intercept is a key point on the graph of a linear equation where the line crosses the y-axis, which corresponds to the value of y when t is zero. In this case, the y-intercept represents the initial condition before any action is taken, or before time starts to change.

For our pool example, the y-intercept is (0, 1000). This means when no water has been pumped out yet (t = 0 minutes), the pool still contains the original 1000 gallons of water. This value is crucial as it serves as the starting value for modeling the situation.

T-Intercept

Alternatively, the t-intercept (also known as the x-intercept in some contexts) indicates the point where the line crosses the t-axis, typically representing the time variable in real-world applications. It is found by setting the y value to zero and solving for t.

In our scenario, the t-intercept is (20, 0), which tells us that after 20 minutes of pumping, there will be no water left in the pool, capturing the moment when the action being modeled—pumping out water—completes its course.