Problem 37 - Precalculus: Mathematical 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, \(\$ 60\) for wheat, and \(\$ 50\) for soybeans. Because of market demand he 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

The farmer should plant 250 acres of corn, 500 acres of wheat, and 450 acres of soybeans.

Step 1: Define Variables

Let's define the 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.

Step 2: Set Up Equations from Constraints

Based on the problem description, we have the following constraints:1. Total land: \( x + y + z = 1200 \).2. The cost constraint: \( 45x + 60y + 50z = 63750 \).3. Market demand constraint: \( y = 2x \) (twice as many acres of wheat as corn).

Step 3: Substitute Market Demand in Equations

Use the equation \( y = 2x \) to eliminate \( y \):Replace \( y \) in the first equation: \( x + 2x + z = 1200 \), simplifying to \( 3x + z = 1200 \).Replace \( y \) in the cost equation: \( 45x + 60(2x) + 50z = 63750 \), simplifying to \( 165x + 50z = 63750 \).

Step 4: Solve the Simplified Equations

We now have:1. \( 3x + z = 1200 \)2. \( 165x + 50z = 63750 \)From the first equation, express \( z \) in terms of \( x \): \( z = 1200 - 3x \).Substitute into the second equation:\[ 165x + 50(1200 - 3x) = 63750 \]Simplify and solve for \( x \):\[ 165x + 60000 - 150x = 63750 \]\[ 15x = 3750 \]\[ x = 250 \]

Step 5: Calculate Acres for Each Crop

With \( x = 250 \):1. Substitute \( x \) back into \( y = 2x \):\( y = 2(250) = 500 \).2. Substitute \( x \) into \( z = 1200 - 3x \):\( z = 1200 - 3(250) = 450 \).

Step 6: Verify All Conditions

Verify that the values satisfy all original constraints:1. Total land: \( 250 + 500 + 450 = 1200 \) (correct).2. Cost: \( 45(250) + 60(500) + 50(450) = 63750 \) (correct).3. Market demand: \( y = 2x \) means \( 500 = 2(250) \) (correct).

Key Concepts

Equations

In solving linear systems, equations are the foundational tools we use to represent relationships between variables. In our problem, we are dealing with three variables: the number of acres for corn (\( x \)), wheat (\( y \)), and soybeans (\( z \)). Each of these crops requires certain acreage, cost constraints, and market conditions, all represented by equations. The first equation, derived from the total land available, is:

• \( x + y + z = 1200 \), which ensures the total land used doesn't exceed 1200 acres.

The second equation represents the cost constraint:

• \( 45x + 60y + 50z = 63750 \), which accounts for the total budget of $63,750 allocated for growing these crops.

The third equation arises from market demand, specifying that the farmer must plant twice as many acres of wheat as corn:

• \( y = 2x \)

These equations are used together to form a system, providing the conditions necessary to solve the problem by finding the values that satisfy all these relationships simultaneously.

Problem Solving

To solve the problem, we use a step-by-step approach to handle the system of equations effectively. Initially, identifying the variables is crucial. Here, we choose \( x \), \( y \), and \( z \) for the acres of corn, wheat, and soybeans respectively.

Next, we build equations based on the constraints given. By substituting the given market demand condition \( y = 2x \) into the land and cost equations, we effectively reduce the complexity of the system. This substitution simplifies the first equation to \( 3x + z = 1200 \), and the cost equation to \( 165x + 50z = 63750 \).

By isolating one variable in the simpler equation, \( z = 1200 - 3x \), we can substitute \( z \) in the second equation. This isolates \( x \), enabling us to solve for its value. Once \( x \) is determined, the other variables \( y \) and \( z \) can be calculated using the relationships specified earlier. This methodical solution strategy involves logical reasoning and algebraic manipulation, crucial skills in problem solving.

Constraints

Constraints in linear systems specify the boundaries within which solutions must be found. They define conditions that any solution needs to satisfy. In our problem, the constraints ensure that the land usage and costs remain within the given limits.

First, the total land constraint \( x + y + z = 1200 \) ensures that the combined acreage for all crops does not surpass the available land. Each variable interaction must respect this limit, preventing overuse.

Secondly, the cost constraint \( 45x + 60y + 50z = 63750 \) keeps the total expenditure for planting within the allocated budget. This constraint ensures financial viability and compliance with monetary conditions.

Finally, the market demand constraint \( y = 2x \) dictates the proportion between wheat and corn based on demand. This type of constraint can often arise from external conditions, like market forces, impacting how resources are allocated.

These constraints together determine a feasible region, focusing the problem on finding solutions that are viable under all given conditions. Balancing these is key in solving real-world problems effectively.