Problem 5
Question
For Activities 1 through \(6, \quad\) for each linear model a. give the slope of the line defined by the equation. b. write the rate of change of the function in a sentence of interpretation. c. evaluate and give a sentence of interpretation for \(f(0)\). The production of a coated wire is \(f(x)=100 x\) feet, wh \(x\) dollars is the amount spent on raw materials.
Step-by-Step Solution
Verified Answer
The slope is 100. Rate of change: 100 feet per dollar. \(f(0) = 0\) means no money spent yields no production.
1Step 1: Identify the Slope
The given equation of the linear function is \(f(x) = 100x\). The general form of a linear equation is \(f(x) = mx + b\), where \(m\) is the slope. By comparing the given equation with this form, we identify that the slope \(m = 100\).
2Step 2: Write the Rate of Change Interpretation
The slope of the function represents the rate of change. For this function, the slope of 100 means that for every dollar spent on raw materials, the production of coated wire increases by 100 feet. This can be interpreted in a sentence as: "The rate of change of the function is 100 feet per dollar spent on raw materials."
3Step 3: Evaluate \(f(0)\)
To find \(f(0)\), substitute \(x = 0\) into the function \(f(x) = 100x\). Thus, \(f(0) = 100 \times 0 = 0\).
4Step 4: Interpret \(f(0)\) with Context
The value \(f(0) = 0\) means that if no money is spent on raw materials, the production of coated wire is 0 feet. This can be interpreted as: "If 0 dollars are spent on raw materials, no wire is produced."
Key Concepts
SlopeRate of ChangeIntercept
Slope
The slope of a linear function is a critical concept in understanding how changes in one quantity affect another. In the general form of a linear equation, denoted as \(f(x) = mx + b\), the term \(m\) represents the slope. The slope is essentially a measure of how steep the line is.
In our specific example with the equation \(f(x) = 100x\), determining the slope is straightforward. By comparing this with the standard linear form, we see that \(m = 100\). Hence, the slope is 100. But what does this mean?
Slope gives us valuable information about the relationship between the variables. In this scenario, for every increase of 1 unit in the amount spent on raw materials (\(x\)), the production of coated wire increases by 100 units (feet). When you visualize this on a graph, the line rises sharply, indicating that even a small increase in \(x\) results in a significant change in the output, \(f(x)\).
Understanding the slope helps us predict and understand the behavior of real-world processes represented by linear equations.
In our specific example with the equation \(f(x) = 100x\), determining the slope is straightforward. By comparing this with the standard linear form, we see that \(m = 100\). Hence, the slope is 100. But what does this mean?
Slope gives us valuable information about the relationship between the variables. In this scenario, for every increase of 1 unit in the amount spent on raw materials (\(x\)), the production of coated wire increases by 100 units (feet). When you visualize this on a graph, the line rises sharply, indicating that even a small increase in \(x\) results in a significant change in the output, \(f(x)\).
Understanding the slope helps us predict and understand the behavior of real-world processes represented by linear equations.
Rate of Change
The rate of change in a linear function quantifies how a change in one variable leads to a change in another variable. In linear functions, this rate is constant and is equivalent to the slope, \(m\). This constant nature makes linear functions predictable and easy to work with.
Focusing on our equation \(f(x) = 100x\), the slope—or the rate of change—is 100. This means the function shows a consistent increase of 100 feet of coated wire for each dollar spent on raw materials. Thus, we can interpret the rate of change as follows:
Focusing on our equation \(f(x) = 100x\), the slope—or the rate of change—is 100. This means the function shows a consistent increase of 100 feet of coated wire for each dollar spent on raw materials. Thus, we can interpret the rate of change as follows:
- For each 1 dollar increase in spending on raw materials, there is a 100 feet increase in wire production.
- This relationship helps in forecasting production levels when budget adjustments are made.
Why is understanding the rate of change important? It allows one to make informed decisions by predicting how alterations in input (money) will influence the output (production). Such insights can be crucial for planning and optimizing resources.
Intercept
The y-intercept in a linear equation, represented in the form \(f(x) = mx + b\), is the value of \(f(x)\) when \(x = 0\). This is where the line crosses the y-axis.
In the case of our equation \(f(x) = 100x\), the intercept, \(b\), is 0. Evaluating \(f(0)\) confirms this: when no money is spent on raw materials (\(x = 0\)), the function yields \(f(0) = 0\).
What does this tell us about our wire production? Simply put, it expresses that with zero expenditure on raw materials, production is non-existent—0 feet of wire are produced. This result reinforces the concept that production is directly reliant on the expenditure of resources.
The intercept provides a baseline or starting point for a linear relationship. Understanding this baseline is crucial because it tells us the initial state of the system being analyzed. Knowing that you start from zero production without investment helps in planning and understanding the necessity of resources in starting up production.
In the case of our equation \(f(x) = 100x\), the intercept, \(b\), is 0. Evaluating \(f(0)\) confirms this: when no money is spent on raw materials (\(x = 0\)), the function yields \(f(0) = 0\).
What does this tell us about our wire production? Simply put, it expresses that with zero expenditure on raw materials, production is non-existent—0 feet of wire are produced. This result reinforces the concept that production is directly reliant on the expenditure of resources.
The intercept provides a baseline or starting point for a linear relationship. Understanding this baseline is crucial because it tells us the initial state of the system being analyzed. Knowing that you start from zero production without investment helps in planning and understanding the necessity of resources in starting up production.
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