Problem 69 - Algebra and Trigonometry

Question

Production Cost 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?

Step-by-Step Solution

Answer

The slope of 6 means each toaster oven adds $6 to the cost, and the y-intercept of $3000 is the fixed production cost.

Step 1: Identify the Components of the Equation

The given equation is \(y = 6x + 3000\). This is a linear equation in the slope-intercept form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Here, \(m = 6\) and \(b = 3000\).

Step 2: Plot the Y-Intercept

To draw the graph, start by plotting the y-intercept. The y-intercept \(b = 3000\) is the point where the line crosses the y-axis. Plot the point \((0, 3000)\) on the graph.

Step 3: Use the Slope to Plot a Second Point

The slope \(m = 6\) is the rise over run, which means for every increase of 1 in \(x\), \(y\) increases by 6. Starting from the y-intercept \((0, 3000)\), move 1 unit to the right (increase \(x\) by 1) and 6 units up to plot another point.

Step 4: Draw the Line

Using the points \((0, 3000)\) and \((1, 3006)\), draw a straight line extending in both directions. Ensure the line is extended such that it can be clearly seen as a linear function.

Step 5: Interpret the Slope

The slope \(m = 6\) represents the rate of change of cost with respect to the number of toaster ovens produced. Specifically, it indicates that for each additional toaster oven produced, the cost increases by $6.

Step 6: Interpret the Y-Intercept

The y-intercept \(b = 3000\) represents the fixed cost of production when no toaster ovens are produced. It is the baseline cost of production, regardless of how many toaster ovens are manufactured.

Key Concepts

Slope-Intercept Form

In mathematics, the slope-intercept form is a way to represent linear equations. It has the standard structure \( y = mx + b \). In this equation, \( m \) represents the slope, and \( b \) is the y-intercept. This format is especially useful because it directly shows two important features of a line:

• The slope, \( m \), which describes how steep the line is.
• The y-intercept, \( b \), which tells where the line crosses the y-axis.

In our example equation, \( y = 6x + 3000 \), you can quickly identify that \( m = 6 \) and \( b = 3000 \). These values are not just numbers; they have specific meanings related to the context, which we will explore further in the next sections.

Slope Interpretation

The slope of a line provides valuable insight into the behavior of a linear equation. Simply put, the slope is a measure of how much \( y \) changes for a certain change in \( x \). In the slope-intercept equation \( y = mx + b \), the slope \( m \) indicates this change. In our given equation \( y = 6x + 3000 \), the slope \( m \) is 6. Here's what this means:

• For each additional toaster oven produced, the cost increases by $6.
• It reflects a consistent rate of change, where the cost will rise steadily as production goes up.
• Visually, this means the line rises 6 units vertically for every 1 unit it moves horizontally on the graph.

Understanding the slope helps predict how changes in production levels will impact overall costs.

Y-Intercept Interpretation

The y-intercept in a linear equation is a vital component as it provides information about the value of \( y \) when \( x \) equals zero. This anchoring point is indicated by \( b \) in the slope-intercept form \( y = mx + b \). In the equation \( y = 6x + 3000 \), \( b = 3000 \), which represents:

• The fixed production cost, even if no toaster ovens are produced.
• This is the starting cost, essentially the base cost incurred before any additional production.
• On a graph, this point is where the line crosses the y-axis, at (0, 3000).

Understanding the y-intercept helps decode the baseline costs involved in production and clarifies the impact of producing additional units on this base.