Problem 65 - Algebra and Trigonometry

Question

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 is 6 (cost per toaster); the \( y \)-intercept is 3000 (fixed cost).

Step 1: Understand the Linear Equation

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

Step 2: Determine the Slope and Y-intercept

From the equation \( y = 6x + 3000 \), the slope \( m \) is 6, and the \( y \)-intercept \( b \) is 3000. This means the cost increases by \(6 for each additional toaster oven produced, and the fixed cost (when no toasters are produced) is \)3000.

Step 3: Plot the Y-intercept on the Graph

To sketch the graph, start by plotting the \( y \)-intercept, (0, 3000), on the vertical axis of a coordinate plane. This point represents the fixed production cost when zero toaster ovens are produced.

Step 4: Use the Slope to Find Another Point

Starting from the \( y \)-intercept (0, 3000), use the slope to find another point. The slope 6 can be interpreted as "rise over run," which means that for every 1 unit increase in \( x \) (toaster ovens), \( y \) (cost) increases by 6 units. Thus, moving right to \( x = 1 \) gives \( y = 3006 \). Plot the point (1, 3006).

Step 5: Draw the Line

Draw a straight line through the points (0, 3000) and (1, 3006). This line represents the production cost as a function of the number of toaster ovens produced.

Step 6: Interpret the Slope

The slope of the line, 6, represents the variable cost of producing one additional toaster oven, i.e., each additional toaster oven increases the production cost by $6.

Step 7: Interpret the Y-intercept

The \( y \)-intercept, 3000, represents the fixed production cost when no toaster ovens are produced. This is the base cost the manufacturer incurs regardless of production level.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is fundamental in mathematics. This form is expressed as \( y = mx + b \), where \( m \) represents the slope and \( b \) is the \( y \)-intercept. In our example equation, \( y = 6x + 3000 \), it is already in slope-intercept form, making it easier to identify the slope and intercept directly from the equation. The beauty of this form lies in its straightforwardness:

• It allows you to quickly determine the slope and \( y \)-intercept.
• It's useful for graphing linear equations because it provides a clear starting point (\( b \)) and directs the steepness and direction of the line (\( m \)).

By following these easy steps, you can interpret and graph any linear equation formatted like this.

Slope Interpretation

The slope is a measure of how steep a line is. In the context of linear equations, the slope shows the rate of change between the two variables on the x-axis and y-axis. In our equation \( y = 6x + 3000 \), the slope is \( m = 6 \). This tells us that for every additional toaster oven produced (which is a 1-unit increase in \( x \)), the cost \( y \) increases by $6. This type of slope represents the variable cost:

• This means the slope indicates changes in cost per unit of item produced.
• A positive slope means costs increase as production increases, while a negative slope would suggest costs decrease with increased production.

Interpreting the slope in terms of cost and production helps us understand how business decisions will impact total spending.

Y-Intercept Interpretation

The \( y \)-intercept in a linear equation is the point where the line crosses the \( y \)-axis. This point gives us valuable information about the equation's real-world context when \( x = 0 \). For the equation \( y = 6x + 3000 \), the \( y \)-intercept is \( b = 3000 \). This represents the fixed production cost when no toaster ovens are manufactured. Fixed costs are constant and do not change with production levels:

• These are costs that must be paid regardless of the level of output.
• This could include expenses like rent, salaries, or machinery.

Understanding the \( y \)-intercept helps businesses recognize their baseline expenses, offering insights into budget management and pricing strategies.