Problem 9 - 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. $$ \begin{array}{l} {C(x)=45 x+300,000} \\ {R(x)=65 x} \end{array} $$

Step-by-Step Solution

Answer

Total profit function: \[ P(x) = 20x - 300,000 \]. Break-even point: \( x = 15,000 \).

Step 1: Understanding the Given Information

Identify the cost function and revenue function given: \[ C(x) = 45x + 300,000 \] \[ R(x) = 65x \]

Step 1 - Finding the Total Profit Function

The total profit function, P(x), is given by the difference between the total revenue function R(x) and the total cost function C(x). \[ P(x) = R(x) - C(x) \]

Step 2 - Calculate P(x)

Substitute the given functions into the profit formula: \[ P(x) = 65x - (45x + 300,000) \] Simplify the equation: \[ P(x) = 65x - 45x - 300,000 \] \[ P(x) = 20x - 300,000 \]

Step 3 - Setting Up the Break-Even Point

The break-even point is where the total profit, P(x), is zero. So, set the profit function equal to zero and solve for x: \[ 20x - 300,000 = 0 \]

Step 4 - Solving for x at the Break-Even Point

Solve the equation from Step 3 to find the break-even point: \[ 20x = 300,000 \] \[ x = \frac{300,000}{20} \] \[ x = 15,000 \]

Key Concepts

profit function

In order to understand the concept of a profit function, we first need to look at what profit means in a business context. Profit is the financial gain obtained when the revenue generated from selling goods or services exceeds the costs associated with producing them.

The profit function, denoted as \(P(x)\), helps us calculate this gain. It is derived from the difference between the total revenue function \(R(x)\) and the total cost function \(C(x)\).

Here's the formula:

\[ P(x) = R(x) - C(x) \]

In the given exercise, the total cost function is \(C(x) = 45x + 300,000\) and the total revenue function is \(R(x) = 65x\). By substituting these into the profit formula, you get:

\[ P(x) = 65x - (45x + 300,000) \]

Simplifying, we have:

\[ P(x) = 20x - 300,000 \]
This equation shows the profit made for any number of units \(x\) produced and sold.

cost and revenue functions

Cost and revenue functions are essential concepts in analyzing a company's financial health.

The total cost function, \(C(x)\), represents the total expenses incurred to produce \(x\) units of a good or service. In our exercise, \(C(x) = 45x + 300,000\) indicates that there are fixed costs of \(300,000 and variable costs of \)45 per unit.

On the other hand, the total revenue function, \(R(x)\), shows the total income from selling \(x\) units. For this example, \(R(x) = 65x\) means each unit fetches $65.

Understanding these functions is vital because they provide insight into the financial dynamics of production and sales. They directly inform how profit can be calculated and when a business will break even.

solving equations

Solving equations is crucial in break-even analysis as it helps identify the point where a company neither makes a profit nor incurs a loss.

To find the break-even point, we set the profit function \(P(x)\) to zero because at this point, total revenue equals total cost:

\[ 20x - 300,000 = 0 \]

Solving this equation involves simple algebraic steps:

1. Add 300,000 to both sides: \[ 20x = 300,000 \]
2. Divide both sides by 20 to find \(x\): \[ x = \frac{300,000}{20} \] \[ x = 15,000 \]

This result tells us that the company must produce and sell 15,000 units to break even. Beyond this point, it starts to make a profit.