Problem 38 - Elementary Algebra Within Reach

Question

In Exercises 35-38, use a graphing calculator to graph the cost and revenue equations in the same viewing window. Find the sales \(x\) necessary to break even \((R=C)\) and the corresponding revenue \(R\) obtained by selling \(x\) units. (Round \(x\) to the nearest whole unit.) $$ C=0.25 x+25,000 \quad R=0.45 x $$

Step-by-Step Solution

Answer

To break even, 125000 units need to be sold, generating a revenue of 56250.

Step 1: Create the Break-Even Equation

Calculate the break-even point, where the cost equals the revenue. This is the solution to the equation \(0.25 x + 25,000 = 0.45 x\).

Step 2: Solve the Equation for x

Rearrange the equation to solve for the x-variable. This involves changing the equation to \(0.45x - 0.25x = 25000\). As a result, the equation becomes \(0.2x = 25000\). The final step is to solve for x by dividing both sides of the equation by 0.2. This gives the result \(x = 125000\).

Step 3: Calculate the Corresponding Revenue

Substitute the x-value from the break-even point into the revenue equation to find the corresponding revenue. By substituting \(x = 125000\) into \(R = 0.45x\), we find that \(R = 0.45 * 125000 = 56250\).

Key Concepts

Using the Graphing Calculator

A graphing calculator is an invaluable tool for visualizing mathematical concepts, especially when dealing with complex functions. In the context of finding the break-even point between cost and revenue, a graphing calculator can offer both numerical solutions and graphical representations. To use it effectively:

• Input the cost and revenue equations as separate functions.
• Adjust the viewing window to accommodate the expected range of values for the break-even point.
• Look for the point where the cost and revenue graphs intersect; this is your break-even point.

In the given exercise, plotting the cost equation, \( C=0.25x+25,000 \), and the revenue equation, \( R=0.45x \), would show the intersection point graphically, making it easier to confirm the calculations.

Cost and Revenue Equations

Understanding cost and revenue equations is fundamental in economics and business. These equations express how the cost to produce goods (cost equation) and the money made from selling those goods (revenue equation) depend on the quantity of goods, \( x \), produced and sold.
The general form of a cost equation is \( C = mx + b \), where \( m \) is the variable cost per unit, \( b \) is the fixed cost, and \( C \) represents the total cost. The revenue equation typically has the form \( R = nx \), where \( R \) is the total revenue and \( n \) is the revenue per unit sold.
In this lesson, the equations were \( C = 0.25x + 25,000 \) and \( R = 0.45x \). The variable costs are \(0.25 and \)0.45 per unit for cost and revenue respectively, while $25,000 represents the fixed costs of production.

Solving for X

To 'solve for x' means to find the value of the variable \( x \) that makes an equation true. In a break-even analysis, this involves setting the cost equal to the revenue and finding the point at which a business neither makes a profit nor suffers a loss.
To solve for \( x \), follow these steps:

• Rearrange the equation to isolate \( x \) on one side.
• Simplify the equation by combining like terms and performing arithmetic operations.
• Use algebraic methods such as division to solve for \( x \).

From our example, when you set \( 0.25x + 25,000 \) equal to \( 0.45x \), and after simplification, you will divide both sides by \( 0.20 \) to find the break-even quantity of \( x = 125,000 \) units.

Corresponding Revenue Calculation

Once the break-even quantity \( x \) is determined, the next step is calculating the corresponding revenue. This involves substituting the break-even quantity into the revenue equation. Here's a simplified process:

• Use the previously calculated break-even point as your \( x \) value in the revenue equation \( R = nx \).
• Multiply this \( x \) value by the unit revenue to calculate total revenue.

In our exercise, substituting the break-even point into the revenue equation, \( R = 0.45x \), and using \( x = 125,000 \), we calculate the corresponding revenue to be \( R = 56,250 \). This represents the revenue at which the business breaks even — not making a profit, but not incurring a loss either.