Problem 83 - Algebra for College Students

Question

Peripheral Visions, Inc., finds that the cost of producing \(x\) studio-quality DVDs is \(C=20 x+100,\) while the revenue produced from them is \(R=24 x(C\) and \(R\) in dollars).

Step-by-Step Solution

Answer

Profit function: \(P(x) = 4x - 100\)

Step 1: Identify the Cost Function

The cost of producing \(x\) studio-quality DVDs is represented by the cost function: \(C = 20x + 100\)

Step 2: Identify the Revenue Function

The revenue from selling \(x\) studio-quality DVDs is represented by the revenue function: \(R = 24x\)

Step 3: Calculate the Profit Function

Profit is calculated by subtracting the cost function from the revenue function: \(P(x) = R - C\)Substitute \(R\) and \(C\): \(P(x) = 24x - (20x + 100)\)

Step 4: Simplify the Profit Function

Distribute and combine like terms: \(P(x) = 24x - 20x - 100$$P(x) = 4x - 100\)

Key Concepts

Profit Function

Understanding the profit function is crucial in determining how much money a company makes after covering all its costs. The profit function is defined as the difference between the revenue and the cost functions. Specifically, it tells us the profit earned when producing and selling a certain quantity of items. The formula for profit function can be expressed as follows:

Let's use our original exercise as an example. Here, the profit function is found by subtracting the cost function from the revenue function:
If the revenue function is given by \( R(x) = 24x \) and the cost function is given by \( C(x) = 20x + 100 \), the profit function will be:
\[ P(x) = R(x) - C(x) \] Substitute the given functions:
\[ P(x) = 24x - (20x + 100) \] Simplify by combining like terms:
\[ P(x) = 4x - 100 \] This resulting profit function tells us that for each DVD produced and sold, the company earns a profit of 4 dollars, but there is also a fixed cost of 100 dollars that needs to be covered first.

In summary:

• Revenue Function \( R(x) \)
• Cost Function \( C(x) \)
• Profit Function \( P(x) \)

Cost Function

The cost function represents the total cost of producing a certain number of items (in this case, studio-quality DVDs). It typically includes both variable costs (which depend on the number of items produced) and fixed costs (which do not change with the number of items).
In our original exercise, the cost function is given as:
\[ C(x) = 20x + 100 \] Here:

• \(20x\) represents the variable cost. For each DVD produced, the cost increases by 20 dollars.
• 100 represents the fixed cost. This is a one-time cost incurred regardless of the number of DVDs produced, possibly due to initial setup costs or overhead.

Understanding and calculating the cost function helps businesses in budgeting and setting up a pricing strategy to ensure profitability.
In summary, the cost function highlights the financial outflow required to produce a specific number of items, balancing between fixed and variable costs.

Revenue Function

The revenue function captures the total income generated from selling a certain number of items. It is essential for assessing how much money a company can make from its sales.
In the initial exercise, the revenue function is represented as:
\[ R(x) = 24x \] Here:

• \