Problem 53 - Algebra Form and Function

Question

The perimeter of a rectangle of length \(l\) and width \(w\) is \(P=2 l+2 w\) (a) Write the expression as a power function of \(l\) or \(w\), if possible. (b) If the length is three times the width, write \(P\) as a power function of \(w\) and give the values of the coefficient \(k\) and the exponent \(p\).

Step-by-Step Solution

Answer

If the length is three times the width, what is the power function of \(w\) with the coefficient \(k\) and the exponent \(p\)? Answer: It is not possible to rewrite the given expression as a power function of either \(l\) or \(w\). However, when the length is three times the width, the perimeter \(P\) as a power function of \(w\) is \(P = 8w\), with the coefficient \(k=8\) and the exponent \(p=1\).

Step 1: Identify the given expression

We have been given the expression for the perimeter of a rectangle which is \(P = 2l + 2w\). Our task is to rewrite this expression as a power function of \(l\) or \(w\) if possible.

Step 2: Analyze the expression

Looking at the given expression, we can see that it is a linear function in terms of both \(l\) and \(w\). A power function has the form \(y = kx^p\), where \(x\) is a single variable, and \(k\) and \(p\) are constants. Unfortunately, we cannot rewrite our given expression as a power function because it is a linear function and contains two different variables, \(l\) and \(w\). So, it's not possible to rewrite it as a power function of either \(l\) or \(w\). Answer (a): It is not possible to rewrite the given expression as a power function of \(l\) or \(w\). (b) If the length is three times the width, write \(P\) as a power function of \(w\), and give the values of the coefficient \(k\) and the exponent \(p\).

Step 1: Apply the given relationship between length and width

We are given that the length is three times the width, meaning \(l = 3w\). Let's substitute this relationship into the perimeter formula: \(P = 2(3w) + 2w\).

Step 2: Simplify the expression

Now, we will simplify the expression by multiplying and combining like terms: \(P = 6w + 2w\) \(P = 8w\)

Step 3: Identify the power function

From the simplified expression, we can see that \(P = 8w\). This is a power function, as it has the form \(y = kx^p\), with \(k=8\) and \(p=1\). Answer (b): The perimeter \(P\) as a power function of \(w\) is \(P = 8w\), with the coefficient \(k=8\) and the exponent \(p=1\).

Key Concepts

Understanding Power Functions

A power function is a special type of mathematical expression. It has a general form of \( y = kx^p \). Here, \( y \) represents the dependent variable, while \( x \) is the independent variable. The coefficient \( k \) is a constant that scales the function, and \( p \) is the exponent that determines the degree or power of \( x \).

The nature of power functions varies depending on the value of \( p \):

• If \( p = 1 \), the function is linear, forming a straight line.
• If \( p = 2 \), it's a quadratic function, shaping a parabola.
• Negative values of \( p \) result in a reciprocal function.

In the context of the exercise, the challenge was to express the perimeter of a rectangle as a power function. Initially, it wasn't possible with two variables \( l \) and \( w \). However, substituting \( l = 3w \) simplified it into a power function of \( w \): \( P = 8w \). Here, \( k = 8 \) and \( p = 1 \), indicating the function is linear.

Exploring Linear Functions

Linear functions are among the simplest forms of functions within algebra. They can be represented as \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. This formula forms a straight line when plotted on a graph.

Linear functions satisfy a straightforward relationship where:

• The output \( y \) changes at a constant rate for any change in \( x \).
• The graph is a straight line, implying consistent slope \( m \).

In the perimeter problem, the expression \( 2l + 2w \) is a linear equation because both terms are first-degree polynomials of \( l \) and \( w \). Once the relationship \( l = 3w \) is applied, the perimeter simply becomes \( P = 8w \). This equation aligns with the principle of linear functions, where \( P \) changes in direct proportion to \( w \).

Decoding Algebraic Expressions

Algebraic expressions are combinations of variables, constants, and operational symbols. They're crucial for translating real-world problems into mathematical models. An expression can:

• Consist of one or more terms, such as \( 2x + 3 \).
• Include operations like addition, subtraction, multiplication, and division.

In our rectangle exercise, the perimeter \( P = 2l + 2w \) is an algebraic expression, succinctly encapsulating the relationship between the length and width of a rectangle. By substituting \( l = 3w \), we converted the expression into \( P = 8w \), simplifying the representation significantly.

This showcases how algebraic expressions help us mold and reshape real-world problems into more manageable mathematical forms, offering clarity and potential solutions to complex issues.