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Chapter 1: Problem 11

For Activities 7 through \(12,\) write a linear model for the given rate of change and initial output value. Fabric sheeting is manufactured on a loom at 4.75 square feet per minute. The first six square feet of the fabric is unusable.

Short Answer

Expert verified
The linear model is \( y = 4.75x - 6 \).

Step by step solution

01

Understand the Problem

We need to create a linear model that represents the production of fabric sheeting over time, considering both the rate of change and the initial output value. The loom manufactures at a rate of 4.75 square feet per minute, but the first 6 square feet are unusable.
02

Identify Key Variables

Let the number of minutes the loom has been running be denoted as \( x \). The total amount of usable fabric produced, in square feet, will be our dependent variable, \( y \).
03

Define the Linear Model Structure

The general structure of a linear model is \( y = mx + b \), where \( m \) represents the rate of change and \( b \) represents the initial value of the dependent variable.
04

Assign Values to Variables

Since 4.75 square feet are produced per minute, \( m = 4.75 \). The first 6 square feet are unusable, so \( b = -6 \) reflects this unusable initial output.
05

Write the Linear Model

Construct the linear equation with \( m = 4.75 \) and \( b = -6 \). The model is \( y = 4.75x - 6 \).
06

Interpret the Model

This model implies that after \( x \) minutes, the usable amount of fabric sheeting produced is \( 4.75x - 6 \) square feet. The \(-6\) accounts for the initially unusable fabric.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rate of Change
In the context of a linear model, the **rate of change** describes how a quantity changes over time. It's a measure of how much the dependent variable increases or decreases for every single unit increase in the independent variable. In our exercise, the loom adds 4.75 square feet of fabric per minute. This means, for each additional minute, there's a consistent increase. The formula part representing this is denoted by the coefficient in front of the variable, which is the rate of production.Understanding rate of change helps in predicting future outcomes, such as how much fabric will be produced after a certain number of minutes.
  • **Constant rate:** In linear models, this rate is constant, meaning it doesn't fluctuate.
  • **Real-life application:** Often used in scenarios like speed, cost, temperature changes over time, etc.
  • **Mathematical representation:** In equations, it's the coefficient 'm' in the equation \(y = mx + b\).
Initial Value Problem
An **initial value problem** in calculus is about finding a function that satisfies a specific condition at a starting point. In linear models, the initial value refers to the starting amount or condition at the origin, which in our exercise, is represented as 6 square feet of unusable fabric.It sets the stage for the equation, indicating the starting functional value before any change over time occurs. The model thus accounts for where the measurement begins by adjusting the starting point with "+6" (usually, it's just \(b\) in equations).
  • **Purpose:** Helps determine where to begin calculations or measurements.
  • **Function attribute:** Sometimes known as the y-intercept in simpler terms for linear equations.
  • **Adjustment concept:** Models must subtract or add this value to adequately reflect true conditions, hence the \(-6\) in \(y = 4.75x - 6\).
Dependent Variables
**Dependent variables** are the outputs or results that depend on one or more other variables, often referred to as the inputs. In calculus, particularly linear modeling, the dependent variable is fundamentally tied to changes in the independent variable.In our example, 'y' is the dependent variable representing the usable amount of fabric produced. As the loom keeps running, 'x', the independent variable, influences how 'y' grows.
  • **Main focus:** In problems, finding how the dependent variable changes is crucial.
  • **Closely linked with:** Rate of change; it shows how the dependent variable will behave.
  • **Ultimate output:** The end goal often involves calculating these dependent variables based on inserted values into the equation. Here, determining the fabric amount via \(y = 4.75x - 6\).
Linear Equations
**Linear equations** in calculus are expressions that create straight lines when graphed. These equations take the form \(y = mx + b\), where:
  • **\(y\):** The dependent variable.
  • **\(m\):** The rate of change or slope, showing steepness or inclination.
  • **\(x\):** The independent variable, often representing time or another regular measure.
  • **\(b\):** The initial value, depicting where the line crosses the y-axis.
In solving scenarios like fabric production, linear equations offer straightforward interpretations and predictions. This model gives an uncomplicated connection between available variables and overarching goals, such as forecasting fabric output at different times.

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