Problem 60 - Algebra and Trigonometry

Question

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 bumper 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 $$ \text { profit }=\text { revenue }-\text { cost } $$ to express \(P(x)\) , the profit on an order of \(x\) stickers, as a difference of two functions of \(x .\)

Step-by-Step Solution

Answer

Profit function: \(P(x) = 0.055x - 0.0000015x^2\)

Step 1: Define Revenue Function

Revenue is defined as the product of the price per sticker and the number of stickers ordered. The price per sticker is given by \(0.15 - 0.000002x\). Hence, the revenue function is the price per sticker multiplied by \(x\), or \(R(x) = x (0.15 - 0.000002x)\).

Step 2: Simplify Revenue Function

Expand the revenue function: \[ R(x) = x (0.15 - 0.000002x) = 0.15x - 0.000002x^2 \] Thus, \(R(x) = 0.15x - 0.000002x^2\).

Step 3: Define Cost Function

The total cost for production is given by \(C(x) = 0.095x - 0.0000005x^2\).

Step 4: Define Profit Function

Profit is defined as revenue minus cost. The profit function \(P(x)\) can be expressed as: \[ P(x) = R(x) - C(x) \]

Key Concepts

Revenue Functions

A revenue function helps determine the total income generated from selling a certain number of goods or services. In the context of the print shop, it involves calculating the total money received from selling bumper stickers. To compute this, you take the number of items sold, multiply it by the price per item, and express it as an algebraic expression.

For example, the price per bumper sticker is given by the expression \(0.15 - 0.000002x\), where \(x\) is the number of stickers ordered. To get the revenue function \(R(x)\), multiply the price per sticker by \(x\):

\[ R(x) = x(0.15 - 0.000002x) \]

This equation simplifies to:

\[ R(x) = 0.15x - 0.000002x^2 \]

Therefore, the revenue function provides a way to express the total earnings based on the number of bumper stickers produced and sold.

Cost Functions

Cost functions represent the total cost involved in producing a specific number of goods. It's an important concept to understand business expenses in relation to production levels. Quite often, cost is expressed as a function of variables, which involve fixed and variable components in real-world scenarios.

For the bumper stickers in the exercise, the cost function \(C(x)\) is straightforwardly given by:

\[ C(x) = 0.095x - 0.0000005x^2 \]

The first part \(0.095x\) signifies the primary cost related to manufacturing, while the term \(-0.0000005x^2\) can account for additional costs as the sticker production increases, reflecting aspects such as bulk manufacturing discounts or other scaling factors.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They are used to represent situations and relationships in a compact form. In business mathematics, algebraic expressions can help model a wide array of scenarios, such as in revenue and cost calculations for our print shop.

In the exercise, both the revenue and cost functions are presented as algebraic expressions.

- The revenue function: \(R(x) = 0.15x - 0.000002x^2\)
- The cost function: \(C(x) = 0.095x - 0.0000005x^2\)

These expressions use variables (\(x\)), and constants with coefficients reflecting real-world dynamics. By defining functions this way, we can manipulate the given expressions algebraically to find other useful functions, such as profit \(P(x)\) which is defined as revenue minus cost. This allows businesses to model and forecast outcomes just by altering the variable \(x\) representing the number of goods being dealt with.