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Chapter 1: Problem 69

A small-appliance manufacturer finds that if he produces \(x\) toaster ovens in a month, his production cost is given by the equation $$y=6 x+3000$$ (where \(y\) is measured in dollars). (a) Sketch a graph of this linear equation. (b) What do the slope and \(y\) -intercept of the graph represent?

Short Answer

Expert verified
The slope shows the cost increase per unit ($6), and the y-intercept shows the fixed cost ($3000).

Step by step solution

01

Identify the Linear Equation

The given equation is \( y = 6x + 3000 \), representing a linear relationship between the number of toaster ovens \( x \) produced and the production cost \( y \).
02

Determine the Slope and Y-intercept

From the equation \( y = 6x + 3000 \), we recognize the slope \( m = 6 \) and the \( y \)-intercept \( b = 3000 \).
03

Interpret the Slope

The slope \( m = 6 \) indicates that for each additional toaster oven produced, the production cost increases by $6.
04

Interpret the Y-intercept

The \( y \)-intercept \( b = 3000 \) represents the fixed production cost per month, which is $3000 regardless of the number of toaster ovens produced.
05

Determine Key Points to Plot

For sketching the graph, identify key points, such as the intercept (0, 3000) and choose a point like when \( x = 1 \), where \( y = 6(1) + 3000 = 3006 \).
06

Sketch the Graph

Plot the identified points on a coordinate system. Start with the \( y \)-intercept at (0, 3000) and another point at (1, 3006). Draw a straight line through these points to form the graph of the equation.

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Key Concepts

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

Slope
The slope in a linear equation is a key concept that indicates how steep a line is on a graph and the direction of that steepness. In the context of a linear equation in the form \(y = mx + b\), the slope is represented by the coefficient \(m\). In our specific equation, \(y = 6x + 3000\), the slope \(m\) is 6. This tells us that for every additional toaster oven produced, the total production cost increases by $6.
  • The sign of the slope indicates direction: a positive slope like our 6 suggests an upward trend as moving from left to right across the graph.
  • If the slope were negative, the line would fall, indicating a decrease in cost with each additional unit produced.
Understanding slope helps determine how changes in one variable affect another, an essential aspect in interpreting and predicting using linear models.
Y-intercept
The \(y\)-intercept of a linear equation is the point where the line crosses the vertical \(y\)-axis. It represents the value of \(y\) when \(x\) is 0. In the formula \(y = mx + b\), \(b\) is the \(y\)-intercept. For our equation \(y = 6x + 3000\), the \(y\)-intercept is 3000.
  • This value signifies the fixed cost of production, which is $3000 in this case, even if no toaster ovens are produced.
  • It provides a starting point for the graph, and illustrates the baseline cost incurred by the manufacturer.
Recognizing and understanding the \(y\)-intercept is crucial for comprehending the fixed elements of a financial analysis using linear equations.
Graphing
Graphing a linear equation visually brings together the individual elements of slope and \(y\)-intercept. To graph the equation \(y = 6x + 3000\), begin by plotting the \(y\)-intercept at the coordinate (0, 3000). This marks where the line meets the \(y\)-axis.
  • The slope determines the next point: from (0, 3000), move up 6 units on the \(y\)-axis and 1 unit to the right on the \(x\)-axis to reach the point (1, 3006).
  • Draw a straight line through these points, extending it across the graph. This line represents all solutions to the equation where the cost relates to the number of toasters produced.
Graphing provides a clear, visual insight into the relationship between variables, helping to predict outcomes and understand trends over a range of values.

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