Problem 21 - Finite Mathematics and Applied Calculus

Question

You manage an ice cream factory that makes two flavors: Creamy Vanilla and Continental Mocha. Into each quart of Creamy Vanilla go 2 eggs and 3 cups of cream. Into each quart of Continental Mocha go 1 egg and 3 cups of cream. You have in stock 500 eggs and 900 cups of cream. You make a profit of \(\$ 3\) on each quart of Creamy Vanilla and \(\$ 2\) on each quart of Continental Mocha. How many quarts of each flavor should you make in order to earn the largest profit?

Step-by-Step Solution

Answer

To maximize profit, the factory should produce 250 quarts of Creamy Vanilla and 0 quarts of Continental Mocha, resulting in a profit of $750.

Step 1: Define the profit function

Let x be the number of quarts of Creamy Vanilla ice cream and y be the number of quarts of Continental Mocha ice cream. The profit function P(x, y) can be expressed as: \[P(x, y) = 3x + 2y\]

Step 2: Identify the constraints

We have constraints on the number of eggs and cream: 1. Eggs constraint: \(2x + y \le 500\) 2. Cream constraint: \(3x + 3y \le 900\) 3. Non-negative production: \(x \ge 0\) and \(y \ge 0\)

Step 3: Simplify the constraints

The cream constraint can be simplified: \(3x + 3y \le 900\) Divide both sides by 3: \(x + y \le 300\)

Step 4: Find the feasible region

The feasible region is the set of all points (x, y) that satisfy all the constraints. Plot the constraints on a graph to find the feasible region.

Step 5: Identify the corner points of the feasible region

There are 3 corner points in the feasible region: 1. (0, 0) 2. (0, 300) 3. (250, 0)

Step 6: Evaluate the profit function at the corner points

1. P(0, 0) = 3(0) + 2(0) = 0 2. P(0, 300) = 3(0) + 2(300) = 600 3. P(250, 0) = 3(250) + 2(0) = 750

Step 7: Determine the optimal solution

The maximum profit is obtained at point (250, 0), where we make a profit of $750. Therefore, to maximize profit, the factory should produce 250 quarts of Creamy Vanilla and 0 quarts of Continental Mocha.

Key Concepts

Profit Optimization

Profit optimization is about maximizing the profitability of a business given specific constraints and available resources. In this context, we manage an ice cream factory that produces two types of ice cream: Creamy Vanilla and Continental Mocha. The goal is to determine how many quarts of each flavor to produce to achieve the highest possible profit.

To start, we define a profit function, which represents potential income based on production levels. Here, this function is given by:

• For Creamy Vanilla: profit per quart is \(3.
• For Continental Mocha: profit per quart is \)2.

This is mathematically expressed as \(P(x, y) = 3x + 2y\), where \(x\) indicates the number of Creamy Vanilla quarts and \(y\) denotes the number of Continental Mocha quarts.

The task is to find values for \(x\) and \(y\) that yield the maximum value for the profit function while considering the given constraints on resources. Evaluating this function at specific feasible production points helps us determine the optimal production level for maximum profit.

Constraint Analysis

Constraint analysis involves evaluating the restrictions and limitations that apply to a decision-making process. These restrictions are often based on resources, time, or other external conditions.

In our ice cream manufacturing example:

• Each quart of Creamy Vanilla requires 2 eggs and 3 cups of cream.
• Each quart of Continental Mocha needs 1 egg and 3 cups of cream.

Given the inventory of 500 eggs and 900 cups of cream, we establish our constraints as follows:

• Eggs: \(2x + y \le 500\)
• Cream: \(3x + 3y \le 900\)

Additionally, we have non-negativity conditions that ensure production is never in deficit: \(x \ge 0\) and \(y \ge 0\). By simplifying and analyzing these constraints, we can identify feasible production levels—those that do not exceed available resources. These constraints form a key part of our strategy to optimize profit, as they define the permissible space within which the profit function can operate.

Feasible Region

The feasible region in linear programming is the set of all possible points that satisfy all the given constraints. It provides a visual representation of potential solutions.

In our scenario, plotting the constraints on a graph helps identify where they intersect or overlap. This creates a polygonal region that highlights feasible options for our production levels.

• From the egg constraint: \(2x + y \le 500\)
• From the cream constraint, simplified to: \(x + y \le 300\)
• Including non-negativity: x and y are both greater than or equal to zero.

The intersection of these conditions forms the feasible region. Key points, or corner points, often yield optimal solutions. In this case, they include (0, 0), (0, 300), and (250, 0).

These corner points are tested in the profit function to find the maximum profit. The feasible region not only assists in understanding what solutions are possible, but also guides us in identifying the most profitable production strategy through the optimization of resources.