Problem 1 - Understanding Elementary Algebra with Geometry

Question

A potter has fixed costs of \(\$ 80 .\) It costs her \(\$ 12\) to produce each piece, and she sells each piece for \(\$ 20 .\) Therefore, her total cost \(C\) for producing \(n\) pieces is given by the equation \(C=80+12 n .\) Her total revenue \(R\) for producing \(n\) pieces is given by the equation \(R=20 n\). (A) Sketch the graph of both equations on the same coordinate system, labeling the horizontal axis \(n\) (B) The potter will break even when her costs and revenue are equal. Use the graph in part (a) to determine the point at which the two lines cross. This is called the break-even point. (C) How many pieces must she sell to break even?

Step-by-Step Solution

Answer

She must sell 10 pieces to break even.

Step 1: Identify the equations

The total cost equation is given by \[C = 80 + 12n\], and the total revenue equation is given by \[R = 20n\].

Step 2: Setup the coordinate system

The horizontal axis represents the number of pieces, \(n\), and the vertical axis represents the cost and revenue in dollars.

Step 3: Plot the Total Cost equation

To plot \[C = 80 + 12n\], choose a few values of \(n\) and calculate \(C\). For example, if \(n = 0\), then \(C = 80\). If \(n = 1\), then \(C = 92\). Plot these points on the graph and draw the line.

Step 4: Plot the Total Revenue equation

To plot \[R = 20n\], choose a few values of \(n\) and calculate \(R\). For example, if \(n = 0\), then \(R = 0\). If \(n = 1\), then \(R = 20\). Plot these points on the graph and draw the line.

Step 5: Determine the break-even point

Find the point where the two lines \(C = 80 + 12n\) and \(R = 20n\) intersect. Set the equations equal to each other: \[80 + 12n = 20n\].

Step 6: Solve for \(n\)

Solve the equation from the previous step: \[80 + 12n = 20n\]. Subtract \(12n\) from both sides: \[80 = 8n\]. Then, divide both sides by 8: \[n = 10\].

Step 7: Interpret the graph

Verify the intersection by checking the graph. The lines intersect at the point where \(n = 10\).

Step 8: State the break-even point

The break-even point is where the total cost equals the total revenue, and from our calculation and graph, this occurs at \(n = 10\).

Key Concepts

fixed costs

Fixed costs are the expenses that do not change regardless of the number of goods produced. In this potter's case, she has fixed costs of \(\text{\textdollar}80\). This amount will stay the same whether she produces zero or a thousand pieces.

Examples of fixed costs include:

• Rent for workshop space
• Insurance
• Salaries of permanent employees

Understanding fixed costs is crucial because they must be covered by your revenue before you can start making a profit.

total cost equation

The total cost equation combines fixed and variable costs to show the complete expense of producing goods. For the potter, the equation is given by \(C = 80 + 12n\).

• \(C\) represents the total cost
• \(80\) is the fixed cost
• \(12n\) is the variable cost, where \(12\) is the cost per piece and \(n\) is the number of pieces

To find the total cost:

• Substitute the number of pieces produced into \(n\)
• Multiply the cost per piece by \(n\)
• Add the fixed cost to get the total cost

For example, if \(n = 5\) pieces, then \(C = 80 + 12(5) = 80 + 60 = 140\). This means it costs \(\text{\textdollar}140\) to produce 5 pieces.

total revenue equation

The total revenue equation shows the income generated from selling goods. For the potter, the equation is \(R = 20n\).

• \(R\) represents the total revenue
• \(20n\) shows that each piece is sold for \(\text{\textdollar}20\)

To calculate total revenue:

• Multiply the selling price per piece by the number of pieces sold \(n\)

For example, if \(n = 5\) pieces are sold, then \(R = 20(5) = 100\). This means the potter makes \(\text{\textdollar}100\) if she sells 5 pieces.

graphing linear equations

Graphing linear equations helps visualize relationships between variables. In this example, we graph the total cost and total revenue equations on the same coordinate system.

Steps to graph both equations:

• Set up the coordinate system with the number of pieces \(n\) on the horizontal axis and cost/revenue on the vertical axis.
• Choose values for \(n\) and calculate corresponding \(C\) and \(R\).
• Plot points for each equation and draw the lines.

For example:

• Plot the cost line \(C = 80 + 12n\) with points like \((0, 80)\) and \((1, 92)\)
• Plot the revenue line \(R = 20n\) with points like \((0, 0)\) and \((1, 20)\)

The point where these lines intersect is the break-even point, which in this case occurs at \((10, 200)\). This means the potter breaks even by selling 10 pieces.