Problem 67 - College Algebra and Calculus: An Applied Approach

Question

The change in volume \(V\) (in milliliters) of the lungs as they expand and contract during a breath can be approximated by the model $$V=\left(-6.549 s^{2}+26.20 s-3.8\right)^{2}, \quad 0 \leq s \leq 4$$ where \(s\) represents the number of seconds. Graph the volume function with a graphing utility and use the trace feature to estimate the number of seconds in which the volume is increasing and in which the volume is decreasing. Find the maximum change in volume between 0 and 4 seconds.

Step-by-Step Solution

Answer

The volume increases in the interval from 0 to approximately 2 seconds and decreases from approximately 2 to 4 seconds. The maximum change in volume can be observed by subtracting the lowest volume from the highest, giving the change in volume between these points.

Step 1: Graphing the function

Use a graphing utility to plot the graph of the function \(V=(-6.549s^{2}+26.20s-3.8)^{2}\) in the interval [0,4].

Step 2: Analyzing the graph

Identify the intervals of time where the graph is rising, that is the volume is increasing, and where it is falling, thus the volume is decreasing. Using the tracing feature can help in this step.

Step 3: Determining the maximum volume change

To estimate the maximum change in volume, analyze the graph function and identify the highest and lowest point of the function within the interval. Subtract the lowest point from the highest point.

Key Concepts

Graphing Utility

The use of a graphing utility is an essential skill for students studying functions in math, particularly in understanding complex relationships like the volume of lungs during breaths. Graphing utilities, such as a graphing calculator or software like Desmos, enable you to visually interpret the behavior of functions over specific intervals. These tools often come with features like zoom, trace, and the ability to plot multiple functions simultaneously, which can provide a more comprehensive understanding of the function's characteristics.

For the exercise given, plotting the volume function on a graphing utility allows you to see the curve that represents how lung volume changes over time. Being able to graph functions is not only key in visual learning but also in identifying specific points of interest, such as where the volume is increasing or decreasing, or where it reaches a peak (maximum volume) or a trough (minimum volume).

Intervals of Increase and Decrease

Understanding the intervals of increase and decrease in a function is important for interpreting its real-life implications. In our lung volume model, identifying these intervals helps to understand when the lungs are filling with air (increasing in volume) and when they are emptying (decreasing in volume).

In mathematical terms, an interval of increase is where the graph of the function moves upward as you go from left to right, meaning the function's output (volume, in this case) is getting larger. Conversely, an interval of decrease is where the graph moves downward. You can use the first derivative, when available, to determine these intervals analytically. However, in many educational settings, graphing utilities with tracing features provide a more intuitive way of finding these intervals, especially for quadratic and higher-order polynomial functions.

Maximum Change in Volume

The concept of maximum change in volume is a critical aspect when examining functions that model physical phenomena. In the lung volume exercise, we are interested in the largest difference between any two volumes within the given time frame. After graphing the function using a graphing utility, you can visually discern the highest and lowest points on the graph.

To find the maximum change, as seen in the step-by-step solution, you would calculate the difference between the maximum and minimum volume values. This represents the peak capacity of the lungs' expansion and contraction in a single breath. In clinical practice, such information could be useful for assessing lung function and health.

Quadratic Functions

Quadratic functions are a fundamental part of algebra and appear in various forms of scientific calculations. They are expressed in the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not zero. When graphed, these functions create a parabolic shape, opening upwards or downwards depending on the sign of \( a \).

In the context of the lung volume problem, we see a quadratic function inside another function, which can be tricky. The volume function is a quadratic expression squared, which means the graph will have a different appearance from a standard parabola. Quadratic functions in such forms require careful analysis as they can have different intervals of increasing and decreasing volumes. A deep understanding of how to manipulate and graph these functions is thus essential for correctly interpreting the changes in lung volume over time.