Problem 36 - Precalculus: Functions and Graphs

Question

A rancher is preparing an oat-cornmeal mixture for livestock. Each ounce of oats provides 4 grams of protein and 18 grams of carbohydrates, and an ounce of commeal provides 3 grams of protein and 24 grams of carbohydrates. How many ounces of each can be used to meet the nutritional goals of 200 grams of protein and 1320 grams of carbohydrates per feeding?

Step-by-Step Solution

Answer

Use 20 ounces of oats and 40 ounces of cornmeal.

Step 1: Identify the Variables

First, we need to identify the variables for our problem. Let \( x \) be the number of ounces of oats, and \( y \) be the number of ounces of cornmeal.

Step 2: Set Up the Equations

Since each ounce of oats provides 4 grams of protein and each ounce of cornmeal provides 3 grams of protein, we set up the equation for protein as \( 4x + 3y = 200 \). For carbohydrates, each ounce of oats provides 18 grams, and each ounce of cornmeal provides 24 grams, giving us the equation \( 18x + 24y = 1320 \).

Step 3: Simplify the Carbohydrates Equation

To make calculations easier, we simplify the carbohydrates equation by dividing the entire equation by 6, resulting in \( 3x + 4y = 220 \).

Step 4: Solve the System of Equations

We now have the system of equations: \( 4x + 3y = 200 \) and \( 3x + 4y = 220 \). We'll use the method of elimination. Multiply the first equation by 3: \( 12x + 9y = 600 \). Multiply the second equation by 4: \( 12x + 16y = 880 \).

Step 5: Eliminate One Variable

Subtract the first new equation from the second: \((12x + 16y) - (12x + 9y) = 880 - 600\) simplifies to \( 7y = 280 \). Solving for \( y \), we get \( y = 40 \).

Step 6: Solve for the Remaining Variable

Substitute \( y = 40 \) back into the first equation: \( 4x + 3(40) = 200 \), which simplifies to \( 4x + 120 = 200 \). Solving for \( x \), we get \( 4x = 80 \), thus \( x = 20 \).

Step 7: Check the Solution

Verify the solution satisfies both original equations. For protein: \( 4(20) + 3(40) = 80 + 120 = 200 \), and for carbohydrates: \( 18(20) + 24(40) = 360 + 960 = 1320 \). Both equations are satisfied.

Key Concepts

Linear Equations

Linear equations are mathematical statements involving variables and constants arranged in a straight line when graphed. In our exercise about oats and cornmeal, linear equations are used to represent how protein and carbohydrate grams add up to the rancher's nutritional goals. Linear equations often take the form \( ax + by = c \), where \( x \) and \( y \) are variables that need to be solved. Here, the nutritional values of oats and cornmeal are represented by these variables which tell us how much of each ingredient is needed to reach the desired protein and carbohydrate levels.

Elimination Method

The elimination method is a handy technique for solving systems of linear equations. Its main goal is to eliminate one of the variables so you can easily solve for the other. In our oats and cornmeal problem, we set up two equations: one for protein and one for carbohydrates. To use the elimination method, first, the equations are adjusted to have the same coefficients for one of the variables. By subtracting or adding these equations, we "eliminate" one variable, allowing us to find the value of the other. This method is particularly effective when dealing with compatible equations like the ones in this example.

Nutritional Goals

In this problem, the nutritional goals refer to the specific targets for grams of protein and carbohydrates. Meeting these goals ensures the livestock is nourished adequately. The objectives are reached by determining the right mix of oats and cornmeal, leveraging their known nutritional content. Each ounce of oats or cornmeal adds a predictable amount of nutrition to the mix, and the goal is to combine them in a way that fulfills the exact amounts needed. By setting up and solving equations based on these known values, we can figure out how much of each ingredient is necessary to satisfy the dietary requirements.

Problem-Solving Steps

The approach to solving this problem follows a structured series of steps.
To begin, variables are defined to clearly represent the unknowns, in this case, the amounts of oats and cornmeal.

• Next, set up linear equations based on the nutritional content.
• Simplify the equations to make calculations more manageable.
• Apply the elimination method to efficiently solve the system of equations.
• Proceed by solving for both variables to ensure the requirements are both satisfied.
• Finally, it's crucial to verify that the solution aligns with the original equations to ensure accuracy and completeness in solution.

Each step builds logically on the previous one to gradually work toward the solution, illustrating a clear problem-solving strategy.