Problem 12 - 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. $$ \begin{array}{l} {C(x)=30 x+49,500} \\ {R(x)=85 x} \end{array} $$

Step-by-Step Solution

Answer

The total-profit function is \( P(x) = 55x - 49,500 \) and the break-even point is at \( x = 900 \).

Step 1 - Identify the Total-Cost and Total Revenue Functions

Given the total-cost function, \( C(x) = 30x + 49,500 \), and the total revenue function, \( R(x) = 85x \).

Step 2 - Find the Total Profit Function

The total profit function, \( P(x) \), is found by subtracting the total-cost function from the total revenue function: \[ P(x) = R(x) - C(x) \] Substitute the given functions: \[ P(x) = 85x - (30x + 49,500) \] Simplify the expression: \[ P(x) = 85x - 30x - 49,500 \] \[ P(x) = 55x - 49,500 \]

Step 3 - Find the Break-even Point

The break-even point occurs when the total profit, \(P(x)\), is zero. Solve the equation \( P(x) = 0 \): \[ 55x - 49,500 = 0 \] Add 49,500 to both sides: \[ 55x = 49,500 \] Divide both sides by 55: \( x = \frac{49,500}{55} \) \( x = 900 \)

Key Concepts

Total-Cost Function

The total-cost function, denoted by \( C(x) \), represents the overall cost to produce \( x \) units of a product or service. In our exercise, the total-cost function is given by \( C(x) = 30x + 49,500\). This function encompasses both variable and fixed costs.
Variable costs change with the level of output. Here, \( 30x \) signifies that it costs \( 30 \) dollars to produce each additional unit.

Fixed costs, on the other hand, do not change with the level of output. They are constant regardless of how many units you produce. In this example, the fixed cost is \$49,500\.
To summarize, the total-cost function provides a complete picture of the expenses needed to produce \( x \) units. It allows you to plan and understand the expense dynamics in your production process.

Total Revenue Function

The total revenue function, denoted by \( R(x) \), represents the total income generated from selling \( x \) units of a product or service. In our exercise, the total-revenue function is given by \( R(x) = 85x \). This means that each unit sold brings in \$85\.
Understanding the total revenue function is crucial for businesses to forecast their sales and income. It helps in pricing strategies and identifying the number of units needed to hit financial targets.
By plotting the total revenue function against the total-cost function, businesses can determine profitability and sustainability. It's essentially the financial heartbeat that keeps businesses motivated and goals clear.

Break-even Point

The break-even point is the level of output at which total revenue equals total costs, leading to zero profit. It can be found by setting the profit function \( P(x) \) to zero.
For our given functions:
\[ C(x) = 30x + 49,500 \] and \[ R(x) = 85x \]
The profit function \( P(x) \) is:
\[ P(x) = 85x - (30x + 49,500) \]
Simplifying, we get:
\[ P(x) = 55x - 49,500 \]
To find the break-even point, set the profit function to zero:
\[ 55x - 49,500 = 0 \]
Solving for \( x \) gives:
\[ 55x = 49,500 \]
\[ x = \frac{49,500}{55} = 900 \]
Therefore, the break-even point is at \( x = 900 \) units. This means the company needs to sell 900 units to cover all its costs. After this point, additional units sold will contribute to profit.
Understanding the break-even point helps businesses make informed decisions regarding pricing, cost management, and sales targets. It is a foundational concept for financial planning and sustainability.