Question

Let \(R\) represent a company's revenue, let \(C\) represent the company's costs, and let \(x\) represent the number of units produced and sold each day. (a) Find the firm's break-even point; that is, find \(x\) so that \(R=C\) (b) Solve the inequality \(R(x)>C(x)\) to find the units that represent a profit for the company. $$ \begin{array}{l} R(x)=8 x \\ C(x)=4.5 x+17,500 \end{array} $$

Short answer

Break-even point: 5000 units. Profit: When more than 5000 units are produced and sold.

Step-by-step solution

Set Equations Equal to Each Other

To find the firm's break-even point, set the revenue equal to the cost: \[ R(x) = C(x) \] So, \[ 8x = 4.5x + 17500 \]

Solve for x

First, isolate the terms involving x on one side of the equation: \[ 8x - 4.5x = 17500 \] Then solve for x: \[ 3.5x = 17500 \] Divide both sides by 3.5: \[ x = \frac{17500}{3.5} = 5000 \] Hence, the break-even point is when 5000 units are produced and sold.

Set Up the Inequality

To find the units that represent a profit, set the revenue greater than the cost: \[ R(x) > C(x) \] Therefore, \[ 8x > 4.5x + 17500 \]

Solve the Inequality

First, isolate the terms involving x: \[ 8x - 4.5x > 17500 \] Simplify: \[ 3.5x > 17500 \] Then solve for x: \[ x > \frac{17500}{3.5} = 5000 \] Hence, the company makes a profit when more than 5000 units are produced and sold.

Key concepts

Revenue Function

The revenue function, denoted as \(R(x)\), represents the total income generated from selling \(x\) units of a product. For this exercise, the revenue function is given by \(R(x) = 8x\). This means that for each unit sold, the company earns \(8. To understand this, imagine every single unit adds \)8 to the total revenue.

For example, if the company sells:

• 10 units, the revenue is \(8 \times 10 = 80\) dollars
• 100 units, the revenue is \(8 \times 100 = 800\) dollars

Remember, the revenue function is crucial for determining how well a company is performing financially.

Cost Function

The cost function, denoted as \(C(x)\), represents the total expenses incurred from producing \(x\) units of a product. For this exercise, the cost function is given by \(C(x) = 4.5x + 17500\). This equation can be broken down into two parts:

• Variable Cost: 4.5x – This represents a cost of \(4.50 per unit produced.
• Fixed Cost: 17500 – This is a constant expense regardless of the number of units produced, such as rent, salaries, or equipment.
  
  Now, let's look at an example:
• For zero units, the cost is \)17500 (just fixed cost)
• For 1000 units, the cost is \(4.5 \times 1000 + 17500 = 22000\) dollars
The cost function helps the company keep track of its expenses and foresee how these change with different production levels.

Solving Inequalities

Solving inequalities is crucial to determine when a company makes a profit, which is the main goal of most businesses. To find profitable production levels, we need to solve the inequality where revenue exceeds costs, \(R(x) > C(x)\).

In this exercise, we have:
\(8x > 4.5x + 17500\)

Steps to solve this:

• Subtract \(4.5x\) from both sides: \(8x - 4.5x > 17500\)
• Simplify: \(3.5x > 17500\)
• Divide both sides by 3.5 to isolate \(x\): \(x > \frac{17500}{3.5}\)
• Calculate: \(x > 5000\)
  

This tells us that the company will make a profit when producing and selling more than 5000 units. Understanding how to solve such inequalities is key in making informed business decisions.