Problem 70 - Intermediate Algebra

Question

The function \(C(x)=200 x+400\) gives the cost for a college to offer \(x\) sections of an introductory class in CPR (cardiopulmonary resuscitation). The function \(R(x)=280 x\) gives the amount of revenue the college brings in when offering \(x\) sections of CPR. a. Find the break-even point (where cost = revenue) by graphing each function on the same coordinate system. b. How many sections does the college need to offer to make a profit on the CPR training course?

Step-by-Step Solution

Answer

5 sections to break even; more than 5 sections for profit.

Step 1: Equate Cost and Revenue

To find the break-even point, where cost equals revenue, we set the equations equal: \(C(x) = R(x)\). So, \(200x + 400 = 280x\).

Step 2: Simplify and Solve for x

Subtract \(200x\) from both sides of the equation to isolate the terms involving \(x\): \(400 = 80x\). Then, solve for \(x\) by dividing both sides by 80: \(x = \frac{400}{80}\).

Step 3: Calculate the Break-even Point

Perform the division to find \(x\): \(x = 5\). This means the college breaks even when they offer 5 sections.

Step 4: Determine Profitability

To make a profit, the number of sections offered, \(x\), must be greater than the break-even point. Therefore, the college needs to offer more than 5 sections to make a profit.

Key Concepts

Cost Function

The cost function is a cornerstone in break-even analysis. In this context, the cost function is represented by the equation \(C(x) = 200x + 400\). This function defines the total cost incurred by a college when it offers \(x\) sections of a CPR course.

Let's break down this function:

• Fixed Costs: The 400 in the function is a fixed cost. These are expenses that do not change with the number of sections offered, such as initial setup costs or possibly administrative fees.
• Variable Costs: The term \(200x\) represents variable costs. These change proportionately with the number of sections offered. For each additional section provided, the cost increases by 200.

Understanding these components allows us to see how different numbers of sections affect the overall cost. As more sections are offered, the influence of variable costs becomes more significant.

Revenue Function

The revenue function depicts how much income the college earns from offering CPR sections. For this problem, the revenue function is \(R(x) = 280x\).

Here’s a closer look:

• Price Per Section: The function suggests that for each section offered, the college earns 280. There is no fixed term here because revenue is entirely dependent on the number of sections.
• Linear Growth: Revenue grows linearly with each additional section, meaning the more sections offered, the higher the revenue.

This linear revenue model helps predict income as you adjust the number of sections. The relationship here is straightforward: more sections equal more revenue, directly impacting the potential for profit.

Graphing Linear Equations

Graphing these functions is essential to visually determine the break-even point. In break-even analysis, we're interested in when the cost equals the revenue, indicating no profit or loss.

To achieve this, plot both functions on the same axis:

• Cost Function \(C(x) = 200x + 400\): Start at \(400\) on the y-axis (fixed cost) and rise by \(200\) for each section along the x-axis.
• Revenue Function \(R(x) = 280x\): Begin at 0 since there's no initial fixed revenue and rise \(280\) for each section.

Where these lines intersect is the break-even point. This graphical analysis not only tells us where costs and revenue align but also provides insight into the rate of growth of each function.

Profitability Analysis

Profitability analysis involves determining how many sections need to be offered to move from breaking even to profiting. The key is to exceed the break-even point.

With our equations:

• Break-even Point: Solving \(200x + 400 = 280x\), we found that \(x = 5\). Thus, with 5 sections, cost equals revenue, and there is no profit.
• Profit Threshold: To earn a profit, the college needs to offer more than 5 sections. Each section beyond this point contributes an additional net income (revenue - cost per section).

Analyzing how costs and revenue change with different numbers of sections helps devise strategies for optimizing profit in educational course offerings.