Problem 109 - A Graphical Approach to Precalculus with Limits

Question

Point The break-even point for a company is the point where costs equal revenues. If both cost and revenue are expressed as linear equations, the break-even point is the solution of a linear system. In each exercise. C represents cost in dollars to produce x items. and \(R\) represents revenue in dollars from the sale of \(x\) items. Use the substitution method to find the break-even point in each case-that is, the point where \(C=R\). Then find the value of \(C\) and \(R\) at that point. \(C=20 x+10,000\) \(R=30 x-11,000\)

Step-by-Step Solution

Answer

The break-even point is 2,100 items with cost and revenue both at $52,000.

Step 1: Set Cost Equal to Revenue

To find the break-even point, equate the cost function \(C\) to the revenue function \(R\) as given in the problem. We have:\[ 20x + 10,000 = 30x - 11,000 \]

Step 2: Solve for x

Rearrange the equation from Step 1 to solve for \(x\). Start by subtracting \(20x\) from both sides:\[ 10,000 = 10x - 11,000 \]Now, add 11,000 to both sides to solve for \(x\):\[ 21,000 = 10x \]Divide both sides by 10:\[ x = 2,100 \]

Step 3: Substitute x into Cost Equation to Find C

Substitute the value of \(x = 2,100\) into the cost equation to find \(C\):\[ C = 20(2,100) + 10,000 \]\[ C = 42,000 + 10,000 \]\[ C = 52,000 \]

Step 4: Substitute x into Revenue Equation to Find R

Substitute the value of \(x = 2,100\) into the revenue equation to find \(R\):\[ R = 30(2,100) - 11,000 \]\[ R = 63,000 - 11,000 \]\[ R = 52,000 \]

Step 5: Verify Break-even Point

The calculated values \(C = 52,000\) and \(R = 52,000\) confirm that at \(x = 2,100\), costs equal revenues, verifying that this is indeed the break-even point.

Key Concepts

Linear Equations

A linear equation is a mathematical expression that models a straight line when plotted on a graph. In simpler terms, it is an equation where each term is either a constant or the product of a constant and a single variable. Linear equations typically take the form:

• Standard form: \( ax + by = c \)
• Slope-intercept form: \( y = mx + b \)

Here, \(a\), \(b\), and \(c\) are constants, \(x\) and \(y\) are variables, \(m\) is the slope of the line, and \(b\) is the y-intercept. Linear equations are a basic tool used to represent various real-world applications, such as calculating cost and revenue, which brings us to our next concept, the cost function.

Cost Function

The cost function represents the total cost of producing a certain number of goods or services. In a linear cost function, we can express the cost as:\[ C = mx + b \]where:

• \(C\) is the total cost.
• \(x\) is the number of items produced.
• \(m\) is the variable cost per item.
• \(b\) is the fixed cost (cost that doesn’t change with the number of items produced).

In our exercise, the cost function is \(C = 20x + 10,000\). This indicates that producing each item costs \(20, and there is a fixed production cost of \)10,000, independent of the quantity produced. Understanding this helps identify how production levels affect overall costs.

Revenue Function

The revenue function represents the total income generated from selling a certain number of goods or services. Similarly structured to the cost function, a linear revenue function can be expressed as:\[ R = px + d \]where:

• \(R\) is the total revenue.
• \(x\) is the number of items sold.
• \(p\) is the price per item.
• \(d\) is any additional revenue (zero in this basic model).

In the problem, the revenue function is given by \(R = 30x - 11,000\), implying that each item brings an income of \(30, and there's a revenue adjustment of -\)11,000. This might represent discounts or reductions factored into the revenue. Knowing how revenue depends on sales is crucial for determining profit or loss.

Substitution Method

The substitution method is a technique used to solve systems of equations, particularly useful when dealing with two linear equations. The method involves substituting one equation into the other to find the value of one of the variables. Follow these simple steps:

• Start by solving one of the equations for one variable in terms of the other.
• Replace that variable in the other equation with its expression.
• Solve the resultant equation to find the value of one variable.
• Substitute back to find the other variable.

In our scenario, we equated the cost and revenue functions to identify the break-even point, where profits are neither lost nor gained. By setting \(20x + 10,000 = 30x - 11,000\), and using substitution to solve, we inferred that the break-even quantity is \(x = 2,100\). This means producing and selling 2,100 items will balance costs and revenues exactly.