Problem 56 - 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 profit \(=\) revenue \(-\) 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

The profit function is \(P(x) = 0.055x - 0.0000015x^2\).

Step 1: Define Revenue Function

Revenue is calculated as the price per sticker multiplied by the number of stickers sold. Given the price per sticker is \(0.15-0.000002x\), the revenue function \(R(x)\) is:\[R(x) = x(0.15 - 0.000002x)\] Simplify the expression to find:\[R(x) = 0.15x - 0.000002x^2\]

Step 2: Define Cost Function

The problem states that the total cost of producing \(x\) stickers is given by the function:\[C(x) = 0.095x - 0.0000005x^2\] No additional simplification is needed for the cost function.

Step 3: Calculate Profit Function

Profit \(P(x)\) is defined as the difference between revenue and cost: \[P(x) = R(x) - C(x)\]Substitute the expressions for \(R(x)\) and \(C(x)\):\[P(x) = (0.15x - 0.000002x^2) - (0.095x - 0.0000005x^2)\]Simplify by distributing the negative and combining like terms:\[P(x) = 0.15x - 0.000002x^2 - 0.095x + 0.0000005x^2\]Combine like terms:\[P(x) = 0.055x - 0.0000015x^2\]

Step 4: Write Final Profit Function

The final expression for the profit function \(P(x)\) as a difference of revenue and cost is:\[P(x) = 0.055x - 0.0000015x^2\]

Key Concepts

Revenue Function

Understanding the revenue function is crucial for determining how sales translate into income. In this exercise, the revenue function calculates the total income from selling bumper stickers. The key here is to multiply the quantity of stickers sold () by the price per sticker.
Given the price per sticker as dependent on the number ordered, it is expressed as:\(0.15 - 0.000002x\). Therefore, the revenue function (\(R(x)\)) becomes:

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

Substituting and simplifying, we make it a quadratic expression:

• \(R(x) = 0.15x - 0.000002x^2\)

This function reflects the decreasing unit price as more stickers are ordered, impacting total revenue as quantity varies.

Cost Function

The cost function indicates how much it costs to produce a specific number of bumper stickers. Knowing production costs helps in determining how profitable the operation can be. Here, the cost function is already given in the exercise with:

• \(C(x) = 0.095x - 0.0000005x^2\)

This equation is also quadratic, showing both linear and non-linear cost components. The term \(0.095x\) represents the consistent cost per sticker, whereas the \(0.0000005x^2\) signifies a small additional cost factor increasing non-linearly with the number of stickers produced. Understanding these elements can assist in evaluating cost efficiency.

Price per Sticker

Price per sticker is a dynamic value in this exercise. Unlike a fixed price scenario, the price here decreases as the order size increases. This adjustable pricing encourages larger orders by providing a discount per additional sticker purchased.
The formula given for the price per sticker is:

• \(0.15 - 0.000002x\)

Where \(x\) is the number of stickers ordered. Initial price starts at $0.15, but diminishes slightly with every additional sticker. This is a classic example of how bulk pricing strategies work to stimulate higher volume sales while impacting the revenue curve.

Profit Calculation

Profit calculation is the cornerstone of understanding financial performance. Profit is derived by subtracting total costs from total revenue. The equation for this exercise is straightforward:

• \(P(x) = R(x) - C(x)\)

By substituting the simplified revenue and cost functions, we obtain:

• \(P(x) = (0.15x - 0.000002x^2) - (0.095x - 0.0000005x^2)\)

Simplifying yields:

• \(P(x) = 0.055x - 0.0000015x^2\)

This expression shows that profitability depends both on the number of stickers sold and the scale of production efficiency. Understanding how these interact can guide strategic business decisions, like how many stickers to produce for optimal profit.