Problem 10 - 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. $$ \begin{array}{l} {C(x)=25 x+270,000} \\ {R(x)=70 x} \end{array} $$

Step-by-Step Solution

Answer

The total-profit function is \( P(x) = 45x - 270,000 \). The break-even point is at \( x = 6,000 \).

Step 1: Identify the given functions

The total-cost function is given by \( C(x) = 25x + 270,000 \). The total-revenue function is given by \( R(x) = 70x \).

Step 2: Calculate the total-profit function

The total-profit function \(P(x)\) is found by subtracting the total-cost function from the total-revenue function: \( P(x) = R(x) - C(x) \). Substituting the given functions, we have:\[ P(x) = 70x - (25x + 270,000) \]Simplifying this, we get:\[ P(x) = 70x - 25x - 270,000 \]\[ P(x) = 45x - 270,000 \]

Step 3: Find the break-even point

The break-even point occurs when the total-profit function \(P(x)\) equals zero. Set \(P(x)\) to zero and solve for \(x\):\[ 45x - 270,000 = 0 \]Move 270,000 to the right side:\[ 45x = 270,000 \]Divide both sides by 45:\[ x = \frac{270,000}{45} \]\[ x = 6,000 \]

Key Concepts

total-cost function

The total-cost function represents the total expenses a company incurs to produce a certain number of units, denoted as 'x'. In our example, the total-cost function is given by
\( C(x) = 25x + 270,000 \).
This equation includes:

• Variable costs: These are costs that change with the level of output. Here, it is represented by \(25x\), where 25 is the variable cost per unit, and 'x' stands for the number of units.


• Fixed costs: These are costs that do not change with the level of output. In the equation, this is represented by 270,000, which is a constant amount the company spends regardless of how many units it produces.
Understanding the total-cost function helps in figuring out how costs increase with production, and how economies of scale might play a role.

total-revenue function

The total-revenue function represents the total income a company makes from selling a certain number of units, denoted as 'x'. For our example, the total-revenue function is
\( R(x) = 70x \).
This equation indicates that:

• The revenue earned per unit is \(70. Hence, for each unit sold, the company earns \)70.
• The more units sold, the higher the total revenue. If 'x' units are sold, the revenue is calculated by multiplying x by 70.

Total revenue is crucial for understanding how much income is generated from sales, and it helps in comparing against costs to find profitability.

total-profit function

The total-profit function shows the profit made after deducting total costs from total revenues for a number of units produced and sold, 'x'. The formula for profit is:
\( P(x) = R(x) - C(x) \).
For our example, substituting the given functions, we get:
\[ P(x) = 70x - (25x + 270,000) \]
Simplifying further, we obtain:
\[ P(x) = 70x - 25x - 270,000 \]\[ P(x) = 45x - 270,000 \]
Thus, the profit function is
\( P(x) = 45x - 270,000 \).
This means that for each unit sold, the company makes a profit of \(45, but it has to cover a fixed cost of \)270,000 first. Profit analysis helps in determining how viable a business is over time.

solving equations

Solving equations is a fundamental skill in finding key values like the break-even point. In this problem, we find the break-even point where total profit equals zero. To find this point, we set the total-profit function to zero and solve for 'x':
\[ P(x) = 45x - 270,000 = 0 \]
Move 270,000 to the right side:
\[ 45x = 270,000 \]
Divide both sides by 45:
\[ x = \frac{270,000}{45} \]
Calculating the result:
\[ x = 6,000 \]
Thus, the break-even point is at 6,000 units. This means that producing and selling 6,000 units will cover all costs, and any units sold beyond this point will result in profit.