Problem 41 - Precalculus: Mathematics for Calculus

Question

A farmer has 1200 acres of land on which he grows corn, wheat, and soybeans. It costs \(\$ 45\) per acre to grow corn, \(\$ 60\) to grow wheat, and \(\$ 50\) to grow soybeans. Because of market demand, the farmer will grow twice as many acres of wheat as of corn. He has allocated \(\$ 63,750\) for the cost of growing his crops. How many acres of each crop should he plant?

Step-by-Step Solution

Answer

Plant 250 acres of corn, 500 acres of wheat, and 450 acres of soybeans.

Step 1: Define Variables

Let \( x \) be the number of acres of corn, \( y \) be the number of acres of wheat, and \( z \) be the number of acres of soybeans. We will form equations based on these variables.

Step 2: Set Up the Acreage Equation

The farmer has 1200 acres in total. Therefore, the equation is:\[ x + y + z = 1200 \]

Step 3: Create the Cost Equation

The total cost can be modeled as:\[ 45x + 60y + 50z = 63750 \]

Step 4: Use Market Demand Constraint

The farmer will grow twice as many acres of wheat as corn, which gives us:\[ y = 2x \]

Step 5: Substitute the Constraint into Acreage Equation

Substitute \( y = 2x \) into the acreage equation:\[ x + 2x + z = 1200 \]Simplify to:\[ 3x + z = 1200 \]

Step 6: Substitute the Constraint into Cost Equation

Substitute \( y = 2x \) into the cost equation:\[ 45x + 60(2x) + 50z = 63750 \]Simplify to:\[ 165x + 50z = 63750 \]

Step 7: Solve the System of Equations

We now solve the system:1. \( 3x + z = 1200 \)2. \( 165x + 50z = 63750 \)First, solve equation (1) for \( z \):\[ z = 1200 - 3x \]Substitute \( z = 1200 - 3x \) into equation (2):\[ 165x + 50(1200 - 3x) = 63750 \]Simplify and solve:\[ 165x + 60000 - 150x = 63750 \]\[ 15x + 60000 = 63750 \]\[ 15x = 3750 \]\[ x = 250 \]

Step 8: Calculate Other Crops

Use \( x = 250 \) in \( y = 2x \):\[ y = 2(250) = 500 \]Substitute \( x = 250 \) into \( z = 1200 - 3x \):\[ z = 1200 - 3(250) = 1200 - 750 = 450 \]

Step 9: Verify Solution

Check all conditions:- Total acres: \( 250 + 500 + 450 = 1200 \)- Cost: \( 45(250) + 60(500) + 50(450) = 11250 + 30000 + 22500 = 63750 \)- Markets demand: \( y = 2x \), \( 500 = 2(250) \). All conditions are satisfied.

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with the same set of variables. In our problem, we have three variables, denoted as \( x \), \( y \), and \( z \), which represent the acres of corn, wheat, and soybeans, respectively. These variables are linked through various conditions that need to be satisfied simultaneously.

The first equation is based on the total acreage. The farmer has 1200 acres of land, which is expressed as:

• \( x + y + z = 1200 \)

The second equation relates to the cost constraints of growing these crops:

• \( 45x + 60y + 50z = 63750 \)

These equations form a system that can be solved together to find specific values for \( x \), \( y \), and \( z \) that satisfy both equations. This is crucial in determining the allocation of each crop on the farmer's land.

Optimization

Optimization in linear programming involves finding the best possible solution under a given set of constraints. In this exercise, the farmer needs to optimize the use of his resources to meet all constraints and achieve his goal of planting the crops efficiently.

The goal is to maximize the use of the 1200 acres and the allotted budget of \( \$63750 \). The constraints provided by the farmer (such as specific budget allocation and twice as much wheat as corn) need to be factored into the optimization process to determine the best solution.

This involves using substitution and algebraic manipulation to reduce the equations to a solvable state. By fixing one variable according to a constraint, others can be expressed in terms of it, leading to the solution that best meets all stated requirements.

Constraints

Constraints are the conditions or limits imposed on the variables in the problem. They define the boundaries within which the solution must lie. In this example, the farmer faces several constraints:

• Total land area: The sum of all acres planted must not exceed 1200 (\( x + y + z = 1200 \)).


• Cost constraint: The total costs for planting the crops should not exceed \( \$63750 \) (\( 45x + 60y + 50z = 63750 \)).


• Market demand: The farmer should plant twice as much wheat as corn (\( y = 2x \)).

Each of these constraints influences the determination of \( x \), \( y \), and \( z \). By addressing all constraints simultaneously, we ensure that the solution is feasible and aligns with the farmer's overall strategy.