Problem 36 - 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.) $$ C=2175 x+85,000 \quad R=3525 x $$

Step-by-Step Solution

Answer

The break-even point (\(x\)) and the corresponding revenue (\(R\)) will differ based on the individual values from the graphing calculator. Remember to round \(x\) to the nearest whole unit.

Step 1: Graph the cost function

The cost function provided is \(C = 2175x + 85000\). This function can be graphed with the help of a graphing calculator. Remember to set \(x\) as the number of units sold and \(C\) as the total cost.

Step 2: Graph the revenue function

The revenue function, given as \(R = 3525x\), can also be graphed using the calculator in the same window as the cost function. Here, \(x\) still represents the number of units sold, while \(R\) represents the total revenue.

Step 3: Find the break-even point

The break-even point is found when the cost is equal to the revenue (\(C = R\)). Visually, it's where the two graphs intersect. To find an accurate numerical approximation, use the 'intersect' function on your graphing calculator. Round \(x\) to the nearest whole unit.

Step 4: Find the Corresponding Revenue

To find the corresponding revenue (\(R\)) at break-even, substitute the \(x\) value found in step 3 into the revenue function \(R = 3525x\).

Key Concepts

Cost Function

In break-even analysis, understanding the cost function is vital. It represents the total expense incurred by a business to produce a certain number of goods. The cost function is typically linear and can be expressed as: \( C = mx + b \), where \( C \) is the total cost, \( x \) is the number of units sold, \( m \) is the variable cost per unit, and \( b \) is the fixed cost.
Using the exercise's example, the cost function is \( C = 2175x + 85000 \).

• \( 2175 \) is the variable cost per unit - the cost changes with each additional unit.
• \( 85000 \) is the fixed cost - the overhead costs that don't change regardless of production.

This linear equation helps determine the total cost associated with different levels of production.

Revenue Function

The revenue function calculates the total income generated from selling a certain number of units. It is often expressed linearly as \( R = px \), where \( R \) is the revenue, \( p \) is the price per unit, and \( x \) is the number of units sold.
For this example, the revenue equation is \( R = 3525x \).

• \( 3525 \) is the price per unit – the consistent amount earned per unit sold.

This straightforward equation provides insights on how revenue increases with each additional unit sold.
Your goal is often to find the value of \( x \) where both the cost and revenue functions are equivalent, meaning there is no profit or loss.

Graphing Calculator

A graphing calculator becomes indispensable in solving break-even problems. This tool allows you to visualize both the cost and revenue functions effectively.
When using a graphing calculator:

• Input the cost function \( C = 2175x + 85000 \) and the revenue function \( R = 3525x \).
• Ensure both functions are on the same graph for simultaneous viewing.
• Identify the intersection of these two lines, which represents the break-even point.
• Utilize the calculator's 'intersect' function for precise results.

Seeing the break-even point visually on a graph helps solidify understanding of where costs and revenues align.

Units Sold

In break-even analysis, 'units sold' refers to the number of items a company sells to cover its costs completely.
The exercise sought to find this number of units where the cost equals revenue, which is achieved at the break-even point. By using both the graphing calculator's functionalities and the formulas, you can determine the exact number of units. Here's the approach:

• Use the 'intersect' function on the graph to find \( x \) when \( C = R \).
• In this example, round \( x \) to the nearest whole number for practicality in business applications.

Knowing the break-even units helps businesses make informed production and sales decisions.