Problem 63 - A Graphical Approach to Precalculus with Limits

Question

Sketch a graph that depicts the amount of water in a 100 -gallon tank. The tank is initially empty and then filled at a rate of 5 gallons per minute. Immediately after it is full, a pump is used to empty the tank at 2 gallons per minute.

Step-by-Step Solution

Answer

The graph is a piecewise linear function with a slope of 5 for 20 minutes and -2 for the next 50 minutes, starting at (0,0) to (20,100) then (20,100) to (70,0).

Step 1: Understand the Problem

We need to sketch a graph that shows the amount of water in the tank over time. The tank is initially empty and fills at 5 gallons per minute until it is full (100 gallons). Then, it is emptied at 2 gallons per minute.

Step 2: Determine Filling Time

Calculate the time it takes to fill the tank. The tank fills at 5 gallons per minute, so for a 100-gallon tank, it takes \( \frac{100}{5} = 20 \) minutes.

Step 3: Plot Filling Portion of the Graph

Draw a line starting at the origin (0,0) representing the tank being empty. This line increases linearly, with time on the x-axis and gallons on the y-axis, reaching the point (20,100) after 20 minutes.

Step 4: Determine Emptying Time

Calculate the time it takes to empty the tank. The tank is emptied at 2 gallons per minute, so to empty 100 gallons, it takes \( \frac{100}{2} = 50 \) minutes.

Step 5: Plot Emptying Portion of the Graph

From the point (20,100), draw a line sloping downwards as the tank empties at 2 gallons per minute. This line should end at the point (70,0) after 50 minutes.

Step 6: Review Graph

The graph should consist of a line increasing from (0,0) to (20,100) (filling phase) followed by a line decreasing from (20,100) to (70,0) (emptying phase). Ensure it is clear and labeled appropriately.

Key Concepts

Linear Functions

Linear functions describe relationships with a constant rate of change between two variables. In this exercise, the relationship between the amount of water in the tank and time is a linear function because water fills and empties the tank at constant rates — 5 gallons per minute and 2 gallons per minute respectively.

To visualize a linear function, imagine a straight line on a graph. This line's equation takes the form of \( y = mx + b \), where \( m \) represents the slope (rate of change) and \( b \) represents the y-intercept (starting value). In our case:

• The filling phase of the tank has the equation \( y = 5x \) because the tank starts empty (\( b = 0 \)) and fills at \( 5 \) gallons per minute.
• Similarly, during the emptying phase, the equation \( y = -2x + 140 \) models the tank emptying at \( 2 \) gallons per minute with a starting point of 100 gallons at 20 minutes (hence adjusting the intercept).

Linear functions are incredibly useful in real-world scenarios where changes occur at a steady rate.

Rate of Change

The rate of change is a key concept in this exercise, representing how quickly one variable changes in relation to another. It is often visualized as the slope of a line in a graph.

During the filling phase, water fills the tank at a rate of 5 gallons per minute, indicating a positive rate of change. The slope here is \( +5 \), representing an increasing amount of water over time.

In contrast, the emptying phase shows a different rate, where water leaves the tank at 2 gallons per minute. This is a negative rate of change with a slope of \( -2 \), showing a decreasing trend. The magnitude of the rate indicates steepness of the slope:

• Larger rates result in steeper lines.
• Smaller rates create gentler slopes.

Understanding these rates helps us predict and calculate time-related events, such as when the tank will be full or empty.

Graph Interpretation

Interpreting graphs is crucial for understanding the relationship between variables. In this exercise, the graph helps illustrate how water amount changes over time in the tank, with time on the x-axis and water volume on the y-axis.

During the first phase, a line rises from (0,0) to (20,100). This represents the tank filling uniformly over 20 minutes. The straight line shows a consistent rate of increase without fluctuations.

Once the tank reaches 100 gallons, a downward sloping line travels from (20,100) to (70,0). This depicts the tank emptying steadily over 50 minutes. The entire graph gives us a piecewise function, where two distinct linear segments describe the whole process.

Each segment provides specific information:

• The rise indicates how quickly the tank fills.
• The fall reveals the rate at which it empties.

By analyzing these transitions, one can easily understand and predict how long such processes will take, aiding in real-world planning and decision-making.