Problem 101 - A Graphical Approach to Precalculus with Limits

Question

Solve each application of openations and composition of functions. Dimensions of a Rectangle Suppose that the length of a rectangle is twice its width. Let \(x\) represent the width of the rectangle. (a) Write a formula for the perimeter \(P\) of the rectangle in terms of \(x\) alone. Then use \(P(x)\) notation to describe it as a function. What type of function is this? (b) Graph the function \(P\) as \(Y_{1}\) found in part (a) in the window \([0,10]\) by \([0,100]\). Locate the point for which \(x=4,\) and explain what \(x\) and \(y\) represent. (c) On the graph of \(P\), locate the point with \(x\) -value 4 . Then sketch a rectangle satisfying the conditions described carlier, and evaluate its perimeter if its width is this \(x\) -value. Use the standard perimeter formula. How does the result compare with the y-value shown on your screen? (d) On the graph of \(P\), find a point with an integer y-value. Interpret the \(x\) - and y-coordinates here.

Step-by-Step Solution

Answer

(a) \(P(x) = 6x\), a linear function. (b) The point (4, 24) shows width 4 and perimeter 24. (c) Matches calculated perimeter. (d) At (5, 30), width is 5, perimeter is 30.

Step 1: Define Relationship Between Length and Width

Given that the length of the rectangle is twice its width, if we let the width be \(x\), then the length will be \(2x\).

Step 2: Write Equation for Perimeter

The perimeter \(P\) of a rectangle with width \(x\) and length \(2x\) is given by:\[P = 2 \times ( ext{Length} + ext{Width}) = 2 \times (2x + x) = 2 \times 3x = 6x\].Thus, \(P(x) = 6x\). This is a linear function, as it is in the form \(y = mx\).

Step 3: Graph the Function

To graph \(P(x) = 6x\), plot the function in the window \([0,10]\) by \([0,100]\). The points will form a straight line from the origin. At \(x = 4\), the y-value (\(P(4)\)) is calculated as:\[P(4) = 6 \times 4 = 24\].Therefore, the point \((4, 24)\) represents a rectangle with width 4 and perimeter 24.

Step 4: Interpret the Graph at x = 4

Locate the point \((4, 24)\) on the graph, which represents where the width \(x\) is 4 and the perimeter \(P(x)\) is 24. This corresponds to a rectangle with width 4 and length \(2 \times 4 = 8\). Calculating the perimeter:\[P = 2 \times (8 + 4) = 24\].This matches the y-value on the graph.

Step 5: Find a Point with Integer y-value

From the equation \(P(x) = 6x\), select an x-value that results in an integer y-value. For example, when \(x = 5\), then \(y = 6 \times 5 = 30\). The point \((5, 30)\) corresponds to a rectangle width of 5 and a perimeter of 30.

Key Concepts

Linear Functions and Their Basics

In mathematics, linear functions are among the simplest and most fundamental types of functions. A linear function can be defined in the form of an equation: \( y = mx + c \), where \( m \) and \( c \) are constants and \( x \) is the variable. The term \( m \) represents the slope, showing how steep the line is. The value \( c \) is the y-intercept, which is where the line crosses the y-axis.

In the context of composite functions and rectangles, we often see these concepts applied. In our example, the perimeter function for a rectangle where the length is twice the width is given by \( P(x) = 6x \). This is also a linear function. Here, \( m = 6 \), representing the rate at which the perimeter grows with respect to the width, and there is no constant term \( c \), indicating that when the width is zero, the perimeter is also zero.

Key points about linear functions include:

• They produce straight-line graphs.
• The slope \( m \) indicates the direction and steepness of the line.
• The function is directly proportional if there is no constant term.

Understanding these properties can help one easily identify and work with linear functions in math problems and real-world applications.

Graphing Functions: Visualizing the Rectangle's Perimeter

Graphing functions is a powerful way of visualizing mathematical relationships. It allows you to see how changes in one variable affect another. To graph a function like the perimeter of a rectangle \( P(x) = 6x \), start by setting up a coordinate system where \( x \) is the width of the rectangle and \( y = P(x) \) is the perimeter.

For our specific function, you can plot points by choosing different widths and calculating their corresponding perimeter using \( P(x) = 6x \). For example, when \( x = 0 \), \( P(0) = 0 \). When \( x = 4 \), \( P(4) = 24 \), and when \( x = 5 \), \( P(5) = 30 \). Connect these points with a straight line, which confirms the linear relationship.

Crucial aspects when graphing functions:

• Always start with known points and calculate the corresponding function values.
• Use these points to draw a straight line for linear functions.
• Graphing helps to interpret and verify results, like matching calculated values with graphical points.

Graphing helps us see patterns and make predictions, vital in understanding how different parts of a problem relate to each other.

Understanding the Perimeter of a Rectangle

The perimeter of a rectangle is a fundamental concept in geometry. It is the total distance around the rectangle, measured by adding together all four sides. For a rectangle with one side measured by \( x \) (width) and the opposite side as \( 2x \) (length), the formula for the perimeter \( P \) is \( 2 \times (\text{width} + \text{length}) \).

Substituting the width and length from our example, the formula becomes \( P = 2 \times (x + 2x) = 2 \times 3x = 6x \), showcasing how linear relationships often appear in geometry.

Key insights about the perimeter of a rectangle include:

• The perimeter increases at a constant rate if one side is constant multiple of another.
• It reflects how the size or area of a rectangle affects its boundary.
• Perimeters can help visualizing the space needed around the structure.

Understanding the perimeter is crucial for applications such as finding the fencing needed for a garden or layout planning, thus making it an essential geometry concept.