Problem 26 - Precalculus Student Solutions Manual 5th

Question

Break-Even Interval Find all intervals where each product will at least break even. The cost to produce \(x\) units of baseball caps is \(C=100 x+6000\), while the revenue is \(R=500 x\)?

Step-by-Step Solution

Answer

The break-even interval is all values of \(x\) such that \(x \geq 15\).

Step 1: Define the Break-Even Point

The break-even point occurs where the cost equals the revenue. Mathematically, when the cost function equals the revenue function: \[ C(x) = R(x) \]

Step 2: Set Up the Equation

Given the cost function \(C(x) = 100x + 6000\) and the revenue function \(R(x) = 500x\), set them equal to find the break-even point: \[ 100x + 6000 = 500x \]

Step 3: Solve for \(x\)

Subtract \(100x\) from both sides of the equation to isolate \(x\): \[ 6000 = 500x - 100x \]Simplify the equation: \[ 6000 = 400x \]Divide both sides by 400: \[ x = \frac{6000}{400} \]\[ x = 15 \]

Step 4: Verify the Break-Even Interval

To ensure the interval where the product at least breaks even, \(x\) must be greater than or equal to 15 units. Hence, the break-even interval is: \[ x \geq 15 \]

Key Concepts

cost function

The cost function describes the total cost of producing a certain number of units. In our case, the cost function is given by: \[ C(x) = 100x + 6000 \] The cost function consists of two parts:

• Variable costs: These depend on the number of units produced. Here, it’s represented by \(100x\).
• Fixed costs: These are costs that do not change with the number of units produced. In this case, it’s the fixed cost of 6000.

Understanding the cost function helps in planning production and budgeting for fixed and variable costs.

revenue function

The revenue function represents the total income from selling a certain number of units. For our scenario, the revenue function is: \[ R(x) = 500x \] This function is simpler because it only depends on the number of units sold, denoted by \(500x\). This means for every unit sold, the revenue increases by 500. Knowing the revenue function helps to understand how sales contribute to income and achieving the break-even point.

solving linear equations

To find the break-even point, we need to solve the linear equation derived by setting the cost function equal to the revenue function: \[ 100x + 6000 = 500x \] Follow these steps:

• Subtract 100x from both sides to start simplifying: \( 6000 = 500x - 100x \).
• Simplify further: \( 6000 = 400x \).
• Divide both sides by 400: \( x = \frac{6000}{400} \).
• Finally, solve for \( x \): \( x = 15 \).

Understanding how to solve linear equations is crucial because it helps determine the exact point where costs and revenues balance out, allowing for effective financial planning and analysis.