Problem 55 - Algebra and Trigonometry

Question

\(55-56\) : Revenue, Cost, and Profit A print shop makes bumper stickers for election campaigns. If \(x\) stickers are ordered (where \(x<10,000\) ), then the price per sticker is \(0.15-0.000002 x\) dollars, and the total cost of producing the order is \(0.095 x-0.0000005 x^{2}\) dollars. Use the fact that revenue \(=\) price per item \(\times\) number of items sold to express \(R(x),\) the revenue from an order of \(x\) stickers, as a product of two functions of \(x .\)

Step-by-Step Solution

Answer

The revenue function is \(R(x) = x(0.15 - 0.000002x)\).

Step 1: Understand the Revenue Formula

The general formula for revenue is given as the product of the price per item and the number of items. Therefore, to express the revenue function, we need to multiply the given price per sticker by the number of stickers ordered.

Step 2: Substitute the Price Function

We have the price per sticker as a function of the number of stickers ordered: \[ P(x) = 0.15 - 0.000002x \]Substitute this expression for price in terms of \(x\) into the revenue formula: \(R(x) = P(x) \times x\).

Step 3: Calculate the Revenue Function

Now, substitute \(P(x)\) into the expression for revenue: \[ R(x) = (0.15 - 0.000002x) \times x \]Distribute \(x\) to both terms within the parentheses:\[ R(x) = 0.15x - 0.000002x^2 \]

Step 4: Express Revenue as a Product of Two Functions

To express the revenue as a product of two functions: consider \( f(x) = x \) and \( g(x) = 0.15 - 0.000002x \).Thus, \[ R(x) = f(x) \times g(x) = x(0.15 - 0.000002x) \]

Key Concepts

Price Function

When determining how much to charge for a product, businesses often use a price function. This function shows the relationship between the quantity of items produced and the cost per item. In our scenario, the price function is given by:

• \(P(x) = 0.15 - 0.000002x\)

Here, \(P(x)\) represents the price per sticker, and \(x\) is the number of stickers produced.
The price function decreases as more stickers are produced.
This is because of the negative coefficient \(-0.000002x\), which indicates a decrease in price by \(0.000002\) dollars for every additional sticker.
By understanding this function, businesses can effectively gauge how price adjustments impact revenue.

Revenue Calculation

Revenue calculation is an essential part of understanding a business's profitability. Revenue is calculated by multiplying the price function by the number of items sold.
In our bumper sticker example:

• Revenue \(R(x)\) is calculated as \( R(x) = P(x) \times x \)

Substituting the price function, \(P(x)\), into this formula, we have:

• \( R(x) = (0.15 - 0.000002x) \times x \)

This equation tells us that revenue is dependent on both the price per sticker and the total number sold.
By understanding this concept, businesses can find the right balance between pricing and production to maximize revenue.

Algebraic Expression

An algebraic expression involves performing mathematical operations to simplify and represent complex relationships. In our example, we start with:

• \(R(x) = (0.15 - 0.000002x) \times x \)

Distributing \(x\) through the expression, we get:

• \(R(x) = 0.15x - 0.000002x^2\)

This step simplifies the way we express revenue by breaking down the individual components:
\(0.15x\) represents the linear part, which shows income based on a fixed price, while \(-0.000002x^2\) reflects the adjustment in revenue as production increases.
Simplified algebraic expressions help in making calculations easier and more readable, which is crucial in both math and business.

Production Cost

The production cost involves calculating all the expenses required to manufacture an item. In the bumper sticker scenario, the cost function is introduced as:

• \( C(x) = 0.095x - 0.0000005x^2 \)

This expression determines how the cost changes with the number of stickers produced.
Here:

• \(0.095x\) represents direct costs, which increase linearly with the number of stickers.
• \(-0.0000005x^2\) represents additional costs that might occur due to complications or efficiencies in production size.

Understanding the production cost function allows businesses to gauge the impact of production volume changes on total costs, which helps in pricing decisions and profitability analysis.