Problem 18 - 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. $$ \begin{array}{l} {C(x)=20 x+120,000} \\ {R(x)=50 x} \end{array} $$

Step-by-Step Solution

Answer

The total-profit function is \[P(x) = 30x - 120,000\]and the break-even point is \[x = 4,000\].

Step 1: Write Down the Cost and Revenue Functions

The total cost function is given as \[C(x) = 20x + 120,000\] and the total revenue function is given as \[R(x) = 50x\].

Step 2: Define the Total-Profit Function

The total-profit function is found by subtracting the total cost function from the total revenue function. \[P(x) = R(x) - C(x)\].

Step 3: Substitute the Functions into the Profit Formula

Substitute the given functions into the profit function formula:\[P(x) = 50x - (20x + 120,000)\].

Step 4: Simplify the Profit Function

Simplify the expression to get the total-profit function:\[P(x) = 50x - 20x - 120,000\]\[P(x) = 30x - 120,000\].

Step 5: Set Up the Break-Even Point Equation

At the break-even point, the profit is zero, so set \[P(x) = 0\].This gives the equation \[0 = 30x - 120,000\].

Step 6: Solve for x

Solve the equation \[0 = 30x - 120,000\] by isolating x:\[30x = 120,000\]\[x = 4,000\].

Key Concepts

Total Cost Function

The total cost function represents the total expenses incurred in producing a certain number of units. It is often composed of fixed costs (constant, regardless of output) and variable costs (which increase with the number of units produced). In the given exercise, the total cost function is \(C(x) = 20x + 120,000\) Here,

• The fixed cost is 120,000. This is the cost that occurs even when no units are produced.
• The variable cost per unit is 20. This means it costs 20 units of currency for each additional unit produced.

Understanding this function helps you identify how costs are structured and how they will change with the production level.

Total Revenue Function

The total revenue function captures the total income generated from selling a certain number of units. It is calculated by multiplying the number of units sold \((x)\) by the price per unit. In the problem, the total revenue function is given as: \(R(x) = 50x\) This equation signifies that for each unit sold, you earn 50 units of currency. Hence,

• If you sell one unit, you earn 50.
• If you sell ten units, you earn 500.

Knowing the total revenue helps predict how much money you can make at different production levels.

Break-Even Point

The break-even point is where total revenue equals total cost. At this point, the profit is zero. To find this point, you set the profit function \(P(x)\) to zero and solve for \(x\). From the exercise, the profit function was simplified to: \ P(x) = 30x - 120,000\ By setting it to zero: \(0 = 30x - 120,000\) Solving for \(x\), \(30x = 120,000\) \(x = 4,000\). This means you need to sell 4,000 units to break even. Understanding the break-even point helps businesses determine the minimum sales quantity necessary to avoid losses.

Profit Function

The profit function shows how much profit (or loss) you can expect at different production levels. It's calculated by subtracting the total cost from the total revenue: \P(x) = R(x) - C(x)\ From the exercise, substituting the given functions: \(P(x) = 50x - (20x + 120,000)\). Simplifying, \(P(x) = 30x - 120,000\). The profit function indicates:

• If you produce zero units, you incur a loss of 120,000 (fixed cost).
• As you produce more units, your profit increases because the revenue from each additional unit (50) is higher than the variable cost per unit (20).

Using the profit function helps in strategic decision-making to enhance profitability.