Problem 17 - Elementary and Intermediate Algebra

Question

For each of the following pairs of total-cost and total revenue functions, find (a) the total-profit function and (b) the break-even point. $$ \begin{aligned} &C(x)=75 x+100,000\\\ &R(x)=125 x \end{aligned} $$

Step-by-Step Solution

Answer

The total profit function is \( P(x) = 50x - 100,000 \), and the break-even point is at \( x = 2,000 \) units.

Step 1: Define Total Cost and Total Revenue Functions

Given total cost function is \( C(x) = 75x + 100,000 \) and the total revenue function is \( R(x) = 125x \).

Step 2: Find the Total Profit Function

The total profit function is found by subtracting the total cost function from the total revenue function. Thus, \( P(x) = R(x) - C(x) \). Substituting the given functions, we get:\( P(x) = 125x - (75x + 100,000) \). Simplify to get \( P(x) = 50x - 100,000 \).

Step 3: Find the Break-Even Point

The break-even point occurs when total profit \( P(x) = 0 \). Set the total profit function equal to zero and solve for \( x \):\( 50x - 100,000 = 0 \)\( 50x = 100,000 \)\( x = \frac{100,000}{50} = 2,000 \).

Key Concepts

Total Cost Function

The total cost function, denoted by \(C(x)\), represents the total costs incurred by a company when producing \(x\) units of a product. This includes both fixed and variable costs. In our exercise, the total cost function is given by: \[ C(x) = 75x + 100,000 \]
In this function:

• \(75x\) is the variable cost, where 75 is the cost per unit of the product.
• \(100,000\) is the fixed cost, incurred regardless of the number of units produced.

Therefore, if you produce zero units, the total cost will still be 100,000. As you produce more units, the variable cost will add to the fixed cost, increasing the total cost. Understanding how costs behave helps businesses set their prices and budget effectively.

Total Revenue Function

The total revenue function, denoted by \(R(x)\), indicates the total income a business earns from selling \(x\) units of a product. It’s calculated by multiplying the price per unit by the number of units sold. In our example, the total revenue function is: \[ R(x) = 125x \]
Here, 125 is the price per unit of the product. If the company sells zero units, the total revenue is zero. As the sales units increase, the total revenue also rises proportionately. This function is crucial for predicting earnings and understanding business performance.

Profit Function

The profit function, represented by \(P(x)\), measures the profitability of producing and selling \(x\) units. It's found by subtracting the total cost function from the total revenue function: \[ P(x) = R(x) - C(x) \]
Using the given functions, we get: \[ P(x) = 125x - (75x + 100,000) \]
Simplifying it: \[ P(x) = 50x - 100,000 \]
This means the company's profit increases by 50 for each additional unit sold, after covering the fixed cost of 100,000. The break-even point is where total profit is zero, i.e., \(P(x) = 0\). Solving for \(x\): \[ 50x - 100,000 = 0 \] \[ 50x = 100,000 \] \[ x = \frac{100,000}{50} = 2,000 \] Hence, the break-even point is at 2,000 units. At this level of production, the total revenue equals the total cost, resulting in zero profit. Understanding the break-even point aids in setting sales targets and financial planning.