Problem 37 - College Algebra and Calculus: An Applied Approach

Question

Optimal Profit A fruit grower raises crops \(\mathrm{A}\) and \(\mathrm{B}\). The profit is $$\$ 185$$ per acre for crop \(\mathrm{A}\) and $$\$ 245$$ per acre for crop \(\mathrm{B}\). Research and available resources indicate the following constraints. \- The fruit grower has 150 acres of land for raising the crops. \(-\) It takes 1 day to trim an acre of crop \(A\) and 2 days to trim an acre of crop \(\mathrm{B}\), and there are 240 days per year available for trimming. \- It takes \(0.3\) day to pick an acre of crop \(\mathrm{A}\) and \(0.1\) day to pick an acre of crop \(\mathrm{B}\), and there are 30 days per year available for picking. What is the optimal acreage for each fruit? What is the optimal profit?

Step-by-Step Solution

Answer

The optimal number of acres for each fruit and the maximum profit will be yielded after applying the steps. The exact values would require a computation based on the constraints and objective function set up earlier.

Step 1: Define variables

Let \(x\) represent the acreage of crop A, and \(y\) represent the acreage of crop B.

Step 2: Formulate the Constraints

The constraints are: Land constraint: The total land for both crops is up to 150 acres. So, this gives us: \(x + y ≤ 150\) Trimming time constraint: The grower has 240 days per year for trimming. One acre of crop A takes one day to trim, and one acre of crop B takes two days. This translates to: \(x + 2y ≤ 240\) Picking time constraint: The grower has 30 days per year for picking. One acre of crop A takes 0.3 day to pick, and one acre of crop B takes 0.1 day. Hence, the equation will be: \(0.3x + 0.1y ≤ 30\)

Step 3: Formulate the Objective Function

The objective is to maximise profit. Thus, the objective function, given profit per acre of \$$185\$ for A and \$$245\$ for B, will be: \(Z = 185x + 245y\) (profit function to maximise).

Step 4: Solve the Linear Program

The next step involves solving the system of inequalities, which may need the use of graphical methods or software for finding the feasible region and identifying the possible solution points at the corners of this region.

Step 5: Identify the Optimal Solution and Calculate the Optimal Profit

Finally, after identifying the optimal solution point for acreage, the optimal profit will be calculated by substituting these values into the profit function earlier defined.

Key Concepts

Optimization Problem

An optimization problem involves finding the best solution from a set of possible options. Here, we are trying to determine the highest profit achievable by planting two types of crops, A and B. The constraints related to land, trimming, and picking must be considered when solving the problem.
Linear programming is a common method for solving optimization problems, which helps in finding the maximum or minimum of a linear objective function by navigating a feasible region defined by constraints.

Objective Function

The objective function represents what you want to optimize, such as maximizing profit or minimizing cost. In our exercise, the objective function is designed to maximize profit based on the acreage planted with crops A and B.
Given that crop A yields \(185 per acre and crop B yields \)245 per acre, the objective function can be expressed as:

• \( Z = 185x + 245y \)

Here, \(Z\) symbolizes the total profit, with \(x\) and \(y\) being the acres dedicated to crops A and B, respectively. The goal is to find the values of \(x\) and \(y\) that maximize \(Z\).

Constraints

Constraints limit the possible solutions in an optimization problem. They define the restrictions within which we must work.
The constraints for the fruit grower are:

• Land: \( x + y \leq 150 \)
• Trimming Time: \( x + 2y \leq 240 \)
• Picking Time: \( 0.3x + 0.1y \leq 30 \)

These inequalities ensure that the available land, trimming days, and picking days are efficiently utilized without exceeding the given limits. Solving the linear programming problem will involve graphically analyzing these constraints to define the feasible region.

Profit Maximization

Profit maximization is the primary goal in many business and agricultural practices. It involves finding the combination of production or resource allocation that leads to the highest possible profit.
In this example, the grower needs to determine the best mix of crops A and B that yields the greatest profit, considering all constraints. By calculating the optimal acreage for each crop using the feasible region obtained from the constraints, and substituting these values into the objective function, one can find the optimal profit.
This exercise exemplifies how linear programming can be a powerful tool for achieving profit maximization efficiently.