Problem 58 - College Algebra

Question

Solve for the desired quantity. A cell phone factory has a cost of production \(C(x)=150 x+10,000\) and a revenue function \(R(x)=200 x\). What is the break-even point?

Step-by-Step Solution

Answer

The break-even point is when 200 units are produced and sold.

Step 1: Understanding the Break-Even Point

The break-even point is where the cost of production equals the revenue. This means the company makes no profit but also incurs no loss. So, we need to find the value of \(x\) for which \(C(x) = R(x)\).

Step 2: Setting the Cost Equal to Revenue

We start by setting the cost function equal to the revenue function. This gives us the equation: \[ 150x + 10,000 = 200x \]

Step 3: Isolating the Variable on One Side

Subtract \(150x\) from both sides of the equation to isolate the variable on one side:\[ 10,000 = 50x \]

Step 4: Solving for the Variable

Now, solve for \(x\) by dividing both sides of the equation by 50:\[ x = \frac{10,000}{50} = 200 \]

Step 5: Interpreting the Result

The solution \(x = 200\) implies that the break-even point occurs when 200 units are produced and sold. At this point, the revenue from selling 200 phones will exactly cover the cost of producing them.

Key Concepts

Cost Function: Understanding Its Role

The cost function represents the total cost incurred in producing a certain number of items, which in this case is cell phones. This function considers both variable and fixed costs. In our example, the cost function is given by \(C(x) = 150x + 10,000\). Here, the term \(150x\) represents the variable cost, which changes with the number of units produced — in this case, each unit produced incurs an additional $150 cost. Meanwhile, \(10,000\) is the fixed cost, which is the cost of production that does not change, regardless of the number of units produced. This could include expenses like rent, salaries, and utility bills that organize the production facilities. Understanding the breakdown in a cost function is crucial for determining profitability and making informed business decisions.

Revenue Function: Calculating the Income

The revenue function defines the total income a company earns from selling its products. It is a straightforward calculation of the number of items sold multiplied by the price of each item. In our scenario, the revenue function is \(R(x) = 200x\). This implies that each unit sold brings in \(200. Therefore, for every phone sold, the company gains \)200 in revenue, directly correlating their income to their sales volume. Analyzing the revenue function helps businesses predict income levels based on expected sales volumes. It is critical when planning strategies to reach break-even points and potential profitability.

Algebraic Equations: Solving for the Unknown

Algebraic equations involve finding unknown variables within mathematical expressions by organizing and simplifying the data represented. They are incredibly useful in many real-world problems, including financial calculations like determining break-even points.In the exercise, the break-even point is found by setting the cost function equal to the revenue function, forming the equation \(150x + 10,000 = 200x\). Solving this equation involves basic algebraic manipulation:

• Starting with setting the cost equal to the revenue to identify where no profit or loss is made.
• Isolating the variable involves manipulating both sides to get terms involving \(x\) on one side of the equation, resulting in the equation \(50x = 10,000\).
• Finally, solving for \(x\) by dividing both sides by 50 gives us \(x = 200\).

This solution signifies that when 200 phones are produced and sold, the total production cost equals the revenue, reaching the break-even point where no net gain or loss occurs.