Problem 72 - Precalculus with Limits

Question

PRODUCTION In Exercises 69-72, a small home business creates muffins, bones, and cookies for dogs. In addition to other ingredients, each muffin requires 2 units of beef, 3 units of chicken, and 2 units of liver. Each bone requires 1 unit of beef, 1 unit of chicken, and 1 unit of liver. Each cookie requires 2 units of beef, 1 unit of chicken, and 1.5 units of liver. Find the numbers of muffins, bones, and cookies that the company can create with the given amounts of ingredients. 1000 units of beef 950 units of chicken 905 units of liver

Step-by-Step Solution

Answer

The number of muffins, bones, and cookies that the company can make depends on the solution to the system of linear equations. This solution will be determined by the constraints of the problem, namely the available amounts of beef, chicken, and liver.

Step 1: Identify the constraints

The constraints for the problem are identified by the available ingredients: \n1000 units of beef\n950 units of chicken\n905 units of liver\nEach muffin, bone, and cookie requires different amounts of these ingredients.

Step 2: Formulate the system of linear equations

The system of equations based on the problem can be represented as:\n\(2m + b + 2c = 1000\) (for beef)\n\(3m + b + c = 950\) (for chicken)\n\(2m + b + 1.5c = 905\) (for liver)\nwhere m represents the number of muffins, b represents the number of bones, and c represents the number of cookies.

Step 3: Solve the system of linear equations

The system of linear equations can be solved by using a method like substitution or elimination. Due to complexity of these equations, it might be easier to solve them using a computer program or a calculator with the necessary functions.

Step 4: Check the solution

The solution obtained needs to be checked by substituting the values of m, b, and c in the original system of equations. Also, since we can't have a negative number of muffins, bones, and cookies, we need to ensure that m, b, c are nonnegative.

Key Concepts

Constraints

In linear systems, a constraint is a limitation or condition that a solution must satisfy. When running a small home business like creating dog treats, the ingredients available act as constraints. Here, the business has limited amounts of beef, chicken, and liver.

• Each ingredient has a specified quantity available.
• For beef, it is 1000 units.
• For chicken, it is 950 units.
• For liver, it is 905 units.

These numbers impose boundaries on how many muffins, bones, and cookies can be produced. By understanding these constraints, we ensure that our solutions stay within the feasible range of production.

Linear Equations

Linear equations are mathematical expressions that describe a straight-line relationship between two or more variables. In our problem, the production quantities of muffins (m), bones (b), and cookies (c) must adhere to the ingredient constraints.
We express these relationships using linear equations:

• For beef: \(2m + b + 2c = 1000\)
• For chicken: \(3m + b + c = 950\)
• For liver: \(2m + b + 1.5c = 905\)

These equations together form a system of linear equations. They represent the different requirements each product (muffins, bones, and cookies) has for the ingredients. Understanding and formulating these linear equations is crucial as it converts real-world constraints into a solvable mathematical problem.

Substitution Method

The substitution method is one way to solve systems of linear equations by solving one equation for a variable and substituting that into another equation. This is particularly effective when equations can be easily manipulated.
Steps to apply the substitution method:

• Solve one of the equations for one variable in terms of the others. For instance, solve for \(b\) in terms of \(m\) and \(c\) from the first equation.
• Substitute this expression into the other equations, replacing \(b\).
• Continue to simplify and substitute until each variable is isolated and solved.

Using substitution, you progressively reduce the number of variables in your equations, making the system easier to solve. However, this method can get cumbersome when dealing with many variables or complex coefficients.

Elimination Method

The elimination method solves linear systems by strategically adding or subtracting equations to eliminate one variable at a time. This method is usually preferred when looking to solve problems with clear, integer-based coefficients.
Here's how the elimination method works:

• Align equations for direct comparison, focusing on one variable to eliminate.
• Multiply equations strategically to get matching coefficients for one variable.
• Add or subtract these equations to remove the targeted variable.
• Repeat for remaining variables until each is isolated and solved.

Elimination method is efficient for systems with clear numerical relationships as it often involves fewer steps. When faced with complex problems, using elimination with computational tools can provide quick and precise solutions.