Problem 51 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Correspond to those in Exercises \(15-27\), Section \(5.1 .\) Use the results of your previous work to help you solve these problems. The Johnson Farm has 500 acres of land allotted for cultivating corn and wheat. The cost of cultivating corn and wheat (including seeds and labor) is $$\$ 42$$ and $$\$ 30$$ per acre, respectively. Jacob Johnson has \(\$ 18,600\) available for cultivating these crops. If he wishes to use all the allotted land and his entire budget for cultivating these two crops, how many acres of each crop should he plant?

Step-by-Step Solution

Answer

Jacob Johnson should plant \(300\) acres of corn and \(200\) acres of wheat to use all the allotted land and his entire budget for cultivating these two crops.

Step 1: Define the variables

Let's denote the number of acres of corn to be planted as \(x\), and the number of acres of wheat as \(y\).

Step 2: Create the system of linear equations

Now, we can represent the problem using two linear equations: 1. Total Acres Constraint: The equation representing the total acres of land allotted for cultivating corn and wheat (500 acres) is: \[ x + y = 500 \] 2. Budget Constraint: The equation representing the available budget for cultivating these crops ($18,600) is: \[ 42x + 30y = 18600 \] So, the system of linear equations is: \[ \begin{cases} x + y = 500 \\ 42x + 30y = 18600 \end{cases} \]

Step 3: Solve the system of linear equations

We can use the substitution method or the elimination method to solve this system. Let's use the elimination method. First, solve the first equation for \(x\) to get: \[ x = 500 - y \] Next, substitute the expression for \(x\) from the first equation into the second equation: \[ 42(500 - y) + 30y = 18600 \] Now, simplify and solve for \(y\): \begin{align*} 21000 - 42y + 30y &= 18600 \\ -12y &= -2400 \\ y &= 200 \end{align*} Now, substitute the value of \(y\) back into the expression for \(x\): \[ x = 500 - 200 = 300 \]

Step 4: Interpret the solution

Jacob Johnson should plant 300 acres of corn and 200 acres of wheat to use all the allotted land and his entire budget for cultivating these two crops.

Key Concepts

Algebra

Algebra serves as the cornerstone for solving problems like the one presented where Jacob Johnson is determining how to allocate land for his crops using a system of linear equations. In algebra, variables such as \( x \) and \( y \) are used to represent unknown quantities. In this scenario, \( x \) represents acres of corn and \( y \) acres of wheat. The task involves setting up equations based on given constraints – the total amount of land available and the budget for cultivation.

Understanding how to manipulate these variables and equations is essential in algebra. The substitution and elimination methods, two fundamental algebraic techniques, allow us to find the values of unknowns that satisfy all given conditions. In the exercise, the elimination method streamlined the process by eliminating one variable to solve for the other, thus exemplifying how algebra is applied to find practical solutions.

Practical Tip:

• Always define your variables clearly before starting.
• Check each equation for possible simplification before solving the system.

Linear Programming

Linear programming is a method used to achieve the best outcome in a mathematical model with linear relationships, which often involves maximization or minimization of a linear objective function, subject to a set of constraints. The problem Jacob faces is a precursor to a linear programming scenario where he might want to not only allocate land and budget but also maximize his profits.

Even though the example provided does not explicitly require finding maximum or minimum values, it lays the groundwork by establishing constraints through linear equations. If Jacob's problem included a profit model, such as a specific profit per acre for corn and wheat, linear programming techniques would then be used to maximize his total profit subject to the same constraints.

Real-World Application:

• In business, linear programming aids in resource allocation to optimize profit, reduce cost, or improve efficiency.

Applied Mathematics

Applied mathematics is the application of mathematical methods by different fields such as science, engineering, business, computer science, and industry. In this context, Jacob is applying mathematical principles to real-world farming problems. The system of linear equations he uses is a fundamental tool in applied mathematics:

Challenges in various disciplines can often be translated into mathematical terms and solved using appropriate methods, such as the system of linear equations seen here. This approach is a direct expression of applied mathematics, where abstract concepts such as algebraic equations have tangible implications on decision-making and resource management.

Advantage in Other Fields:

• Understanding such applications can be beneficial in sectors like economics, environmental planning, and even technology design.