Problem 15 - Precalculus: Functions and Graphs

Question

Planning crop acreage \(A\) farmer, in the business of growing fodder for livestock, has 90 acres available for planting alfalfa and com. The cost of seed per acre is $$ 32\( for alfalfa and 48 dollars for corn. The total cost of labor will amount to 60 dollars per acre for alfalfa and $$ 30\) per acre for corn. The expected revenue (before costs are subtracted) is 500 dollars per acre from alfalfa and 700 dollars per acre from com. If the farmer does not wish to spend more than 3840 dollars for seed and 4200 dollars for labor, how many acres of each crop should be planted to obtain the maximum profit?

Step-by-Step Solution

Answer

Plant 30 acres of alfalfa and 60 acres of corn for maximum profit.

Step 1: Define the Variables

Let \( x \) represent the number of acres planted with alfalfa, and let \( y \) represent the number of acres planted with corn. We will create equations based on the problem statement using these variables.

Step 2: Set Up the Constraints

We have several constraints to consider:1. The total acreage constraint: \( x + y \leq 90 \).2. The seed cost constraint: \( 32x + 48y \leq 3840 \).3. The labor cost constraint: \( 60x + 30y \leq 4200 \).These inequalities represent the constraints for the farmer.

Step 3: Write the Objective Function

The objective is to maximize profit, calculated as revenue minus costs. The revenue from alfalfa is \( 500x \), and from corn is \( 700y \). The cost for seed and labor for each is \( 32x + 60x = 92x \) for alfalfa and \( 48y + 30y = 78y \) for corn.The objective function for profit \( P \) is:\[ P = (500x + 700y) - (32x + 60x) - (48y + 30y) = 408x + 622y \]

Step 4: Graph the Constraints

Graph the inequalities derived from the constraints on a coordinate plane. The feasible region is determined by the intersection of these inequalities.- \( x + y \leq 90 \)- \( 32x + 48y \leq 3840 \)- \( 60x + 30y \leq 4200 \)Plot these lines and identify the feasible region where all constraints overlap.

Step 5: Identify Corner Points

The optimal solution lies at one of the feasible region's vertices. Calculate the intersection points:1. Intersection of \( x + y = 90 \) and \( 32x + 48y = 3840 \).2. Intersection of \( x + y = 90 \) and \( 60x + 30y = 4200 \).3. Intersection of \( 32x + 48y = 3840 \) and \( 60x + 30y = 4200 \).

Step 6: Calculate Profit at Each Corner Point

Substitute the coordinates of each corner point into the profit function \( P = 408x + 622y \) to find the profit at each vertex. Compare these values to find the maximum profit.

Step 7: Determine the Optimal Number of Acres

Based on the calculations, identify which combination of \( x \) and \( y \) yields the highest profit. This will be the number of acres of alfalfa and corn that should be planted.

Key Concepts

Objective Function

The objective function is a key concept in linear programming. It's a mathematical expression that you need to maximize or minimize. In business contexts, this often represents cost, revenue, or profit.
For the given problem, we want to find the planting strategy that maximizes profit. The profit is calculated as revenue minus costs. So, for acres of alfalfa and corn, the profit function is defined as follows:

• Revenue from alfalfa is represented by the term \(500x\), where \(x\) is the number of acres of alfalfa.
• Revenue from corn is represented by \(700y\), where \(y\) is the number of acres of corn.
• The cost for seeds and labor results in the terms \(92x\) for alfalfa and \(78y\) for corn.

Putting it all together, our objective function \(P\) aiming to maximize profit is given by:\[ P = (500x + 700y) - (92x + 78y) = 408x + 622y \] This clearly indicates how each additional acre of either crop affects the overall profit.

Constraints in Optimization

In optimization, constraints are the limitations or requirements that must be considered in the solution. These are often represented by inequalities.
In this problem, the farmer has several key constraints to work within:

• The total acreage constraint ensures that the total land used does not exceed available land. It's expressed as the equation \(x + y \leq 90\).
• The seed cost constraint limits the amount spent on seeds, represented by \(32x + 48y \leq 3840\).
• The labor cost constraint is given by \(60x + 30y \leq 4200\), reflecting the maximum affordable labor cost.

Each constraint forms a boundary in the possible combinations of acres for alfalfa and corn. Together, they define the feasible solutions to the problem, ensuring the farmer operates within the available resources.

Feasible Region

The feasible region in linear programming is the area where all constraints of the problem overlap. It represents all possible solutions that satisfy every constraint simultaneously.
To visualize this, you graph all the linear inequalities from your constraints. Each inequality splits the plane into two halves, and the feasible region is the section where all these conditions meet.
For this particular problem, the region is defined by the intersection of the constraints:

• \(x + y \leq 90\) (total acres)
• \(32x + 48y \leq 3840\) (seed cost)
• \(60x + 30y \leq 4200\) (labor cost)

This overlapped area is crucial because the optimal solution, which maximizes profit, resides within this area. Any point outside doesn't satisfy all constraints, thus is infeasible.

Corner Point Method

The corner point method is a systematic way to find the optimum value of the objective function. In linear programming problems, like this one, the solution is found at a vertex of the feasible region.
This method involves:

• Identifying all the vertices (or corner points) of the feasible region by finding the intersections of the boundary lines formed by constraints.
• Calculating the value of the objective function at each of these vertices.

For the exercise, the corner points are identified through the intersection of the constraints:

• Intersection of \(x + y = 90\) and \(32x + 48y = 3840\)
• Intersection of \(x + y = 90\) and \(60x + 30y = 4200\)
• Intersection of \(32x + 48y = 3840\) and \(60x + 30y = 4200\)

By evaluating the objective function at each corner point, you can determine which point provides the maximum profit. This makes it an effective method for finding optimal solutions in problems with linear constraints.