Problem 20 - Algebra A Combined Function

Question

The total revenues (in dollars) for MCD, Inc., Manufacturing Company to sell \(x\) blank audiocassette tapes per week is given by the polynomial function \(R(x)=2 x .\) Find the total revenue from selling 20,000 tapes per week.

Step-by-Step Solution

Answer

The total revenue is $40,000.

Step 1: Understanding the Revenue Function

The revenue function given is \( R(x) = 2x \), where \( x \) represents the number of audiocassette tapes sold. Since we need to find the revenue when 20,000 tapes are sold, \( x = 20,000 \).

Step 2: Substitute the Value of x

We substitute \( x = 20,000 \) into the revenue function. This gives us: \[ R(20000) = 2 \times 20000. \]

Step 3: Perform the Multiplication

Calculate the revenue by performing the multiplication: \[ 2 \times 20000 = 40000. \]

Step 4: Interpret the Result

The calculation shows that the total revenue for selling 20,000 tapes is \( 40,000 \) dollars. This is how much MCD, Inc. will earn per week by selling that quantity of tapes.

Key Concepts

Revenue Calculation

Revenue calculation is integral for businesses to estimate their earnings based on sales. In our example, we have a polynomial revenue function, specified as \( R(x) = 2x \). This formula allows MCD, Inc. to determine their total revenue depending on the number of audiocassettes sold. Here, \( x \) denotes the quantity of tapes sold.

To calculate revenue, companies simply need to identify how many products they’ve sold, which in this instance is represented by \( x \), and then substitute this into their revenue function. This method provides a clear understanding of income corresponding to various sales levels.

When dealing with revenue calculations, remember:

• The coefficient (in this case, 2) reflects earnings per unit sold.
• The function \( R(x) \) changes based on the value of \( x \), offering dynamic revenue predictions.
• Accurate calculations enable strategic business decisions, like sales goals and budgeting.

Substitution in Algebra

Substitution in algebra is a fundamental skill that helps in simplifying expressions and equations. It allows us to solve problems efficiently by replacing variables with their known values. For the given revenue function \( R(x) = 2x \), we substitute \( x \) with a specific number of audiocassettes sold to find the actual revenue.

In the example provided, we know that 20,000 tapes are sold, so we substitute \( x = 20,000 \). This substitution transforms the abstract equation into a solvable arithmetic problem:

\[ R(20000) = 2 \times 20000. \]

Substitution is crucial because:

• It turns theoretical equations into practical calculations.
• It helps verify predictions with actual data points.
• It’s a widely used technique across various areas of mathematics and science.

Mastery of this concept will empower you to tackle real-world problems with confidence and precision.

Multiplication in Algebra

Multiplication in algebra can be seen as taking a repeated sum of a number. In the context of our revenue function, we are multiplying the coefficient, which indicates earnings per item, by the quantity sold. For instance, for our polynomial function \( R(x) = 2x \), after substituting \( x = 20,000 \), the operation becomes:

\[ 2 \times 20000 = 40000. \]

This straightforward operation yields the total revenue.

Some key points about multiplication in algebra include:

• It simplifies large sums into manageable calculations.
• Serves as a fundamental operation for more complex algebraic manipulations.
• Understanding multiplication helps with broader mathematical problem-solving skills.

By practicing multiplication within algebraic contexts, you'll enhance your ability to handle larger data sets and problem scenarios efficiently.