Problem 35 - Algebra 2

Question

The Yoder Family Dairy produces at most 200 gallons of skim and whole milk each day for delivery to large bakeries and restaurants. Regular customers require at least 15 gallons of skim and 21 gallons of whole milk each day. If the profit on a gallon of skim milk is \(\$ 0.82\) and the profit on a gallon of whole milk is \(\$ 0.75,\) how many gallons of each type of milk should the dairy produce each day to maximize profits?

Step-by-Step Solution

Answer

Produce 179 gallons of skim milk and 21 gallons of whole milk each day to maximize profit.

Step 1: Define the Variables

Let \( x \) represent the number of gallons of skim milk the dairy produces each day, and \( y \) represent the number of gallons of whole milk. Our goal is to maximize profit given some constraints.

Step 2: Formulate the Objective Function

The profit from skim milk is \( 0.82x \) and from whole milk is \( 0.75y \). Therefore, the objective function to maximize is \( P = 0.82x + 0.75y \).

Step 3: Identify the Constraints

The constraints based on the problem's requirements are: 1. \( x + y \leq 200 \) (since the total production cannot exceed 200 gallons) 2. \( x \geq 15 \) (at least 15 gallons of skim milk must be produced)3. \( y \geq 21 \) (at least 21 gallons of whole milk must be produced)4. \( x \geq 0 \) and \( y \geq 0 \) (as negative production is not possible).

Step 4: Graph the Constraints

Plot the inequalities on a graph to identify the feasible region. The feasible region is bounded by the lines: \( x + y = 200 \), \( x = 15 \), \( y = 21 \), \( x = 0 \), and \( y = 0 \). The solution must lie in this region.

Step 5: Determine the Corner Points

The corner points of the feasible region are found by solving the equations of the boundary lines. They are:1. Intersection of \( x + y = 200 \) and \( y = 21 \), giving \( (179, 21) \).2. Intersection of \( x + y = 200 \) and \( x = 15 \), giving \( (15, 185) \).3. Point \( (15, 21) \), where \( x = 15 \) and \( y = 21 \).

Step 6: Calculate the Profit at Each Corner Point

Evaluate the profit function \( P = 0.82x + 0.75y \) at each corner point:1. At \( (179, 21) \), \( P = 0.82(179) + 0.75(21) = 146.78 + 15.75 = 162.53 \).2. At \( (15, 185) \), \( P = 0.82(15) + 0.75(185) = 12.3 + 138.75 = 151.05 \).3. At \( (15, 21) \), \( P = 0.82(15) + 0.75(21) = 12.3 + 15.75 = 28.05 \).

Step 7: Identify the Maximum Profit

The maximum profit is at the corner point \( (179, 21) \) and amounts to \( 162.53 \). Thus, the Yoder Family Dairy should produce 179 gallons of skim milk and 21 gallons of whole milk each day to achieve maximum profit.

Key Concepts

Objective Function

In linear programming, the objective function is a crucial concept because it defines what we are trying to achieve. For the Yoder Family Dairy problem, our objective is to maximize profit. The profit is calculated based on the number of gallons of skim and whole milk sold. This is represented by the equation \( P = 0.82x + 0.75y \), where \( x \) is the gallons of skim milk and \( y \) is the gallons of whole milk.
Each term in the objective function represents the profit from each type of milk.
It is important to correctly formulate the objective function because this equation is used to evaluate different potential solutions.
We will analyze specific points to find out which one provides the highest value for profit, aligning with our goal to maximize it.
Remember, the coefficients \(0.82\) and \(0.75\) come from the profit earned per gallon of milk sold.

• Maximizing means looking for the highest possible value of \( P \).
• The objective function equation dictates the decision-making process in finding the best mix of milk production.

Constraints

Constraints are like rules that we must follow in linear programming. They define the limits within which the solution must fall. They include inequalities that restrict the possible values of our variables, \( x \) and \( y \).
For the Yoder Dairy problem, the constraints are:

• \( x + y \leq 200 \): Total milk production cannot exceed 200 gallons each day.
• \( x \geq 15 \): At least 15 gallons of skim milk must be produced.
• \( y \geq 21 \): At least 21 gallons of whole milk must be produced.
• \( x \geq 0 \) and \( y \geq 0 \): Negative production of milk is not possible.

Constraints ensure that every solution we consider is realistic and feasible.
By graphing these constraints, we can visually identify the feasible region that meets all conditions.
Understanding constraints helps in recognizing the importance of limitations while optimizing an objective function in real-world scenarios.
Often, these constraints reflect real-life restrictions such as resource availability, minimum requirements, or maximum capacities.

Feasible Region

The feasible region is a visual representation of all possible solutions that satisfy the problem's constraints. This region is identified by plotting the constraints on a graph.
In this problem, the feasible region is bounded by the lines:

• \( x + y = 200 \)
• \( x = 15 \)
• \( y = 21 \)

It represents the area where all the specified conditions and constraints of the linear programming problem are met.
The corner points or vertices of this region are critical because they are potential candidates for maximizing the objective function. The solution that offers the maximum profit will be located at one of these points.
Visualizing the feasible region helps you see all potential solutions quickly, and from there, you can identify the best one to achieve the desired outcome.
The trick is to check the objective function at each corner of this region, as often, the optimum solution in linear programming lies at a vertice.

Maximization Problem

A maximization problem in linear programming focuses on increasing a specific outcome to its maximum potential. In the context of the Yoder Family Dairy problem, the goal is to produce skim and whole milk in such a combination that the total profit is maximized.
Maximization requires examining the feasible region for the highest value of the objective function, \( P = 0.82x + 0.75y \).
After defining the feasible region, every point in this area satisfies all the constraints, but only certain points (usually the vertices) will also maximize the desired outcome.
The reason we target vertices is that, in linear programming, extreme points often hold the key to the optimal solution.
The methodology of solving a maximization problem involves:

• Defining the objective function you need to maximize.
• Setting up the constraints and drawing your feasible region.
• Evaluating the objective function at each corner point of the feasible region.
• Choosing the point with the highest objective value as the solution.

This structured approach ensures that you efficiently reach the best possible outcome given your starting conditions and constraints.