Problem 61 - A Graphical Approach to College Algebra

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 rises linearly from 0 to 100 gallons in 20 minutes, then falls back to 0 by 70 minutes.

Step 1: Understanding the Problem

We need to trace the amount of water in the tank over time. Initially, the tank is empty and filled at a rate of 5 gallons per minute until it's full. Then, it is emptied at 2 gallons per minute.

Step 2: Determine Filling Time

To fill a 100-gallon tank at 5 gallons per minute, it will take \[\frac{100 \text{ gallons}}{5 \text{ gallons per minute}} = 20 \text{ minutes}\]. Thus, the tank will be full at 20 minutes.

Step 3: Determine Emptying Time

Once the tank is full, the emptying process begins. To empty at 2 gallons per minute, it will take \[\frac{100 \text{ gallons}}{2 \text{ gallons per minute}} = 50 \text{ minutes}\]. Thus, the tank will be empty again at 70 minutes (20 minutes to fill and 50 more to empty).

Step 4: Graph Shape Analysis

The graph starts at 0 gallons. From 0 to 20 minutes, draw a line with a slope of 5, indicating the tank fills linearly to 100 gallons. From 20 to 70 minutes, the slope changes to -2, moving from 100 gallons back to 0.

Step 5: Draw the Graph

On the x-axis, represent the time in minutes, and on the y-axis, represent the gallons of water. First, draw a line from the origin (0,0) to the point (20,100). Next, draw another line from (20,100) to (70,0). The graph forms a piecewise linear function with two segments.

Key Concepts

Piecewise Functions

Piecewise functions are functions defined by different formulas or expressions over different intervals. In the context of this exercise, we are dealing with a piecewise linear function. This function illustrates how the amount of water in a tank changes over time.
To understand piecewise functions, picture a graph broken into different segments, each segment having its own rule. In the water tank example, the function has two segments:

• The first segment shows the tank being filled. It starts at 0 and ends at 20 minutes, increasing at a rate of 5 gallons per minute.
• The second segment depicts the tank being emptied, starting at 20 minutes and ending at 70 minutes. Here, the water level decreases at a rate of 2 gallons per minute.

By using piecewise functions, you can easily model real-world situations with distinct phases or states, like in this tank filling and emptying scenario.

Rate of Change

The rate of change describes how one quantity changes relative to another. Mathematically, it is represented by the slope of a line on a graph. In linear functions, this represents how steep a line is.
In our water tank problem:

• The filling rate is 5 gallons per minute. Graphically, this means each minute of time results in an increase of 5 gallons of water, giving the line a positive slope of 5.
• Once the tank is full, the emptying rate is 2 gallons per minute. Graphically, this implies each minute results in a decrease of 2 gallons of water, resulting in a negative slope of -2.

Understanding the rate of change is vital as it helps predict future behavior and make informed decisions, such as scheduling the filling and emptying times in this scenario.

Real-World Applications

Real-world applications of graphing linear functions and understanding rate of change are vast. These concepts are used in fields like economics, engineering, and everyday problem-solving.
Consider our tank example, which is a perfect illustration of how businesses and industries might manage resources. For instance:

• In the logistics industry, understanding how quickly a tank fills or empties can help design schedules for transporting liquids efficiently.
• In agriculture, farmers might use similar calculations to understand irrigation needs and ensure crops receive the correct amount of water without wastage.
• In the financial world, analyzing rates of change can help track profits over time or optimize investment strategies.

The ability to graph these functions and interpret the rates of change is crucial in optimizing processes and improving efficiencies in various domains of life.