Problem 106 - A Graphical Approach to College Algebra

Question

When a thermal inversion layer is over a city, pollutants cannot rise vertically, but are trapped below the layer and must disperse horizontally. Assume that a factory smokestack begins emitting a pollutant at 8 A.M. and that the pollutant disperses horizontally over a circular area. Let \(t\) represent the time in hours since the factory began emitting pollutants \((t=0\) represents 8 A.M.) and assume that the radius of the circle of pollution is $$r(t)=2 t \quad \text{miles}.$$ The area of a circle of radius \(r\) is represented by $$\mathscr{A}(r)=\pi r^{2}$$ (a) Find \((\mathscr{A} \circ r)(t)\) (b) Interpret \((\mathscr{A} \circ r)(t)\) (c) What is the area of the circular region covered by the layer at noon? (d) Support your result graphically.

Step-by-Step Solution

Answer

(a) \((\mathscr{A} \circ r)(t) = 4\pi t^2\); (b) Area grows quadratically over time; (c) Area at noon: \(64\pi\) square miles.

Step 1: Understanding Composition

The function offered, \(\mathscr{A}(r) = \pi r^2\), calculates the area of a circle with radius \(r\). Recognizing \(\mathscr{A}\) and \(r(t)\): we're viewing a composed function \((\mathscr{A} \circ r)(t)\), which is equivalent to \(\mathscr{A}(r(t))\). This interpretation implies taking the area function \(\mathscr{A}(r)\) and plugging in \(r(t)\) for \(r\).

Step 2: Evaluating the Composite Function

Substitute \(r(t) = 2t\) into \(\mathscr{A}(r) = \pi r^2\). So, \(\mathscr{A}(r(t)) = \pi (2t)^2 = \pi \cdot 4t^2 = 4\pi t^2\). Hence, \((\mathscr{A} \circ r)(t) = 4\pi t^2\).

Step 3: Interpret the Function

The function \((\mathscr{A} \circ r)(t) = 4\pi t^2\) translates to representing the area of the pollutant spread that depends on time contingency. As each hour passes, the area of horizontal dispersion expands quadratically, proportional to \(t^2\).

Step 4: Calculate Area at Noon

Noon stands for \(t = 4\) hours past 8 A.M. Substituting \(t = 4\) into \(4\pi t^2\) gives \(4\pi (4)^2 = 4\pi \times 16 = 64\pi\) square miles.

Step 5: Graphical Representation

Using a graph, plot the function \(\mathscr{A}(t) = 4\pi t^2\) on the y-axis versus \(t\) on the x-axis. This portrays a quadratic curve starting at the origin that ascends steadily, marking off significant points, i.e., \(t = 1, 2, 3, 4\) which helps visually check and support calculations.

Key Concepts

Area of a Circle

The area of a circle is calculated using the formula \( \mathscr{A}(r) = \pi r^2 \), where \( r \) denotes the radius of the circle.
This formula comes from the geometric idea of covering a circular region with infinite small squares, or pi times the radius squared. It demonstrates how much flat space, or two-dimensional area, the circle occupies.
When dealing with circular pollutants like in our scenario, understanding how to determine the area they cover is important.
Key points about circle areas include:

• Depends on the square of the radius.
• Larger radii result in exponentially larger areas.
• In real-world applications, like pollution spread, calculating precise areas helps in estimating impacts.

Radius of a Circle

The radius of a circle is simply the distance from the center point to any point on the circle outline.
It's important as it fundamentally determines the size of the circle and any area calculations.
In our problem, the radius function \( r(t) = 2t \) changes over time because the pollutants disperse.
Let's explore some significance of radius:

• The radius determines both the size and scale of a circular region.
• Since radius can be a function of another variable, like time \( t \) here, it describes dynamic changes.
• Gives context to how quickly a phenomenon, like pollution, spreads over time.

In short, a changing radius can turn a static picture into a dynamic process, illustrating real-world impacts.

Function Interpretation

Function interpretation involves understanding what a function represents and how changes in input (like time) affect output (like area).
For the composite function \((\mathscr{A} \circ r)(t) = 4\pi t^2\), it shows how the pollutant spread area (output) changes as time \( t \) (input) increases.
Here's more about reading and interpreting functions:

• Composite functions like \((\mathscr{A} \circ r)(t)\) involve plugging one function into another.
• They provide a powerful tool for modeling complex processes by connecting various phenomena.
• Understanding these let us predict how different aspects influence each other over time.

Such interpretations are crucial in environmental science in predicting impact areas of pollution or other spreading phenomena.

Graphical Representation

Graphical representations of functions, like plotting \( \mathscr{A}(t) = 4\pi t^2 \), provide a visual way to understand the relationship between variables.
By plotting the function on an axis, you can immediately see how changing one variable affects another.
Let's consider the graphical aspects in detail:

• The graph typically starts at the origin and rises, forming a curve, due to the quadratic nature.
• As \( t \) increases, the curve shows how steeply and quickly the area grows.
• Graphs can pinpoint specific values, confirming calculations, such as \( 64\pi \) at \( t = 4 \) hours.

Thus, visual tools are indispensable for checking mathematical results and better grasping changing processes.