Problem 78 - Algebra and Trigonometry

Question

Maximizing Profit A community bird-watching society makes and sells simple bird feeders to raise money for its conservation activities. The materials for each feeder cost \(6, and the society sells an average of 20 per week at a price of \)10 each. The society has been considering raising the price, so it conducts a survey and finds that for every dollar increase, it loses 2 sales per week. (a) Find a function that models weekly profit in terms of price per feeder. (b) What price should the society charge for each feeder to maximize profits? What is the maximum weekly profit?

Step-by-Step Solution

Answer

The optimal price is $12 per feeder, with a maximum weekly profit of $88.

Step 1: Define Variables

Let \( x \) be the number of dollar increases from the initial price of $10 per feeder. The new price per feeder is then \( 10 + x \).

Step 2: Determine Sales Function

According to the survey, for each dollar increase, the society loses 2 sales per week. The initial sales are 20 feeders, so the new sales count is given by the expression \( 20 - 2x \).

Step 3: Derive Profit Function

Profit is calculated as revenue minus cost. Revenue is the product of price per feeder and the number of feeders sold, \((10 + x)(20 - 2x)\). The total cost is the number of feeders sold times $6, \(6(20 - 2x)\). Hence, the profit function \( P(x) \) is given by:\[ P(x) = (10 + x)(20 - 2x) - 6(20 - 2x) \]

Step 4: Simplify the Profit Function

First expand and simplify the expression for profit:\[ P(x) = 200 + 20x - 20x - 2x^2 - 12x \] Combine like terms:\[ P(x) = -2x^2 + 8x + 80 \]

Step 5: Maximize the Profit

To find the maximum profit, we will find the vertex of the parabola defined by \( P(x) = -2x^2 + 8x + 80 \). The vertex \( x \)-coordinate is given by \(-\frac{b}{2a}\), where \( a = -2 \) and \( b = 8 \).\[ x = -\frac{8}{2(-2)} = 2 \]

Step 6: Calculate Optimal Price and Maximum Profit

Substitute \( x = 2 \) into the price equation \( 10 + x \), so the optimal price is \( 10 + 2 = 12 \). Substitute \( x = 2 \) into the profit function to find the maximum profit:\[ P(2) = -2(2)^2 + 8(2) + 80 = -8 + 16 + 80 = 88 \]

Key Concepts

Profit Function

The profit function is a key concept in maximizing profits through algebra. It allows us to express the total profit made from selling items based on different factors. In this scenario with the bird-watching society, the goal is to find out how changes in price affect overall profit.

Here's how it works:

• First, establish the initial costs and selling price in your scenario. The materials cost \(6 per feeder, and they sell for \)10 each.
• Next, find the changes in your sales linked to price changes. Here, for every dollar increase, 2 fewer sales happen.
• The profit function, therefore, captures these relationships, with the formula
  \[ P(x) = (10 + x)(20 - 2x) - 6(20 - 2x) \]

This formula helps simplify your calculations and figure out total profit based on different price changes. You can see how the product and revenues interplay through algebraic expressions.

Revenue Calculation

Calculating revenue is a crucial part of finding profit. Revenue is the total income from sales before subtracting costs. In the bird-watching society's problem, revenue changes based on both the number of items sold and the selling price.

To calculate revenue:

• Identify the selling price, which is $10 initially, and changes as per the formula to \( 10 + x \).
• Determine the number of sales with changes in price, shown by the equation \( 20 - 2x \).
• Multiply these factors: \( ext{Revenue} = ( ext{New Price})( ext{Number of Feeders Sold}) \).

For example, when the price is increased to \( 10 + x \), sales drop to \( 20 - 2x \). Therefore, the revenue is calculated as \( (10 + x)(20 - 2x) \). This helps us understand how different price changes can impact sales and, consequently, revenue, leading to different levels of profit.

Parabola Vertex

The vertex of a parabola is crucial in maximizing or minimizing functions like the profit function. Since profit is modeled by a quadratic function, its graph is a parabola. The vertex helps identify the point of maximum profit.

To find the vertex:

• Start with the quadratic profit function \( P(x) = -2x^2 + 8x + 80 \).
• The formula to find the vertex is \( -\frac{b}{2a} \), with \( a = -2 \) and \( b = 8 \).
• Apply the formula: \( x = -\frac{8}{2(-2)} = 2 \).

In our case, \( x = 2 \) gives us the price change that maximizes profit. The vertex reveals this optimal point on the curve, allowing the bird-watching society to find the best price for maximum profit.

Optimal Pricing

Optimal pricing is about finding that price point where profit is maximized. The bird-watching society, by adjusting the price, can discover the best possible price for selling their feeders.

To find the optimal price:

• Use the vertex calculation from the parabola equation to find the optimum price change \( x = 2 \).
• Calculate the new optimal price by adding \( x \) to the initial price: \( 10 + 2 = 12 \).

This optimal price point of $12 per feeder is where the society maximizes its weekly profits, as shown in the solution. By finding this balance, the society ensures that even with fewer sales, the profit remains the highest possible given their constraints.