Problem 19 - Algebra A Combined Function

Question

The total revenues (in dollars) for an art supply company to sell \(x\) boxes of colored pencils per week over the Internet is given by the polynomial function \(R(x)=11 x\). Find the total revenue from selling 1500 boxes of colored pencils.

Step-by-Step Solution

Answer

The total revenue is 16500 dollars.

Step 1: Understand the Function

The revenue function given is \( R(x) = 11x \), which implies that the revenue is calculated by multiplying the number of boxes sold, \( x \), by 11 dollars per box.

Step 2: Substitute the Value

Substitute \( x = 1500 \) into the revenue function \( R(x) = 11x \). This will give us the total revenue for selling 1500 boxes.

Step 3: Calculate the Revenue

Now calculate the revenue by multiplying: \( R(1500) = 11 \times 1500 = 16500 \).

Step 4: Conclusion

The total revenue from selling 1500 boxes of colored pencils is \( 16500 \) dollars.

Key Concepts

Revenue Calculation

In revenue calculation, we determine the total income generated from selling a certain number of products or services. In this case, the company sells colored pencils. The revenue function given by the exercise is: \[ R(x) = 11x \]This polynomial function tells us that each box of colored pencils sells for \(11. For any given number of boxes, we multiply that number by 11 dollars to find the total revenue.

• Revenue Function: conceptualized as \( R(x) = ext{price per box} imes ext{number of boxes} \)
• The unit price (here \)11) is a constant that affects the total revenue linearly.
• To calculate the revenue for any value of \( x \), simply plug in the number of boxes into the function.

For example, if we need the revenue from selling 1500 boxes, we substitute 1500 into the function. The calculation requires multiplying 1500 by 11 to find the total revenue, which comes to $16,500.

Mathematical Modeling

Mathematical modeling involves using mathematical equations or functions to represent real-world scenarios. This allows businesses and individuals to predict outcomes and make informed decisions based on mathematical analysis. In this exercise, the revenue function \( R(x) = 11x \) models the relationship between the number of boxes sold and the total revenue generated. The power of using a simple linear model like this lies in its straightforwardness—allowing quick calculations and predictions.

• Identify relevant variables: number of boxes (\( x \)) and revenue (\( R(x) \)).
• Define relationships using equations: here, an equation connecting sales to revenue.
• Solve problems by applying specific numbers: substituting values such as 1500 into \( R(x) = 11x \).

With mathematical models, businesses can also forecast revenue for different sales levels, helping in budgeting and scaling operations based on predictable patterns.

Substitution Method

The substitution method is a powerful technique where a specific value is plugged into an expression or function to make it easier to solve. This method was used in the exercise to calculate the revenue from selling 1500 boxes of colored pencils.In the exercise, we substitute the specific value of 1500 into the revenue function:\[ R(x) = 11x \]Follow these simple substitution steps:

• Identify the function and understand what it represents.
• Choose the correct value to substitute, based on the problem's requirements.
• Replace the variable \( x \) with the chosen value: \( R(1500) \).
• Simplify the expression to find the result.

Substitution is invaluable in scenarios where we must find specific outcomes quickly, like total revenue for a set quantity in business calculations.