Problem 48 - Precalculus: Mathematical for Calculus

Question

A cat food manufacturer uses fish and beef by-products. The fish contains 12 g of protein and 3 g of fat per ounce. The beef contains 6 g of protein and 9 g of fat per ounce. Each can of cat food must contain at least \(60 \mathrm{g}\) \- of protein and \(45 \mathrm{g}\) of fat. Find a system of inequalities that -describes the possible number of ounces of fish and beef that can be used in each can to satisfy these minimum requirements. Graph the solution set.

Step-by-Step Solution

Answer

The system is \( 2x + y \geq 10 \), \( x + 3y \geq 15 \), \( x \geq 0 \), \( y \geq 0 \). The solution set is graph where these inequalities overlap.

Step 1: Define Variables

Let \( x \) represent the number of ounces of fish used in each can, and \( y \) represent the ounces of beef used in each can.

Step 2: Construct Protein Inequality

Since fish provides 12 grams of protein per ounce and beef provides 6 grams of protein per ounce, the total protein can be represented as \( 12x + 6y \). The can must contain at least 60 grams of protein, so the inequality is \( 12x + 6y \geq 60 \).

Step 3: Construct Fat Inequality

Fish provides 3 grams of fat per ounce and beef provides 9 grams of fat per ounce. The total fat is represented as \( 3x + 9y \). The can must contain at least 45 grams of fat, leading to the inequality \( 3x + 9y \geq 45 \).

Step 4: Create Non-negativity Constraints

To ensure realistic solutions, both \( x \) (ounces of fish) and \( y \) (ounces of beef) must be non-negative. Thus, we add the inequalities \( x \geq 0 \) and \( y \geq 0 \).

Step 5: Simplify Inequalities

Simplify the inequality from Step 2: \( 12x + 6y \geq 60 \) becomes \( 2x + y \geq 10 \). Simplify the inequality from Step 3: \( 3x + 9y \geq 45 \) becomes \( x + 3y \geq 15 \).

Step 6: Graph the System of Inequalities

Graph the inequalities on the coordinate plane. The area that satisfies all inequalities (\( 2x + y \geq 10 \), \( x + 3y \geq 15 \), \( x \geq 0 \), \( y \geq 0 \)) is the solution set.

Key Concepts

Linear Programming

Linear programming is a mathematical technique used for optimization. This method helps you find the best outcome, such as maximum profit or minimum cost, under given constraints. In the context of the cat food manufacturer problem, linear programming involves determining how many ounces of fish and beef should be used to meet protein and fat requirements at a minimum level.
In this scenario, the constraints are given by the inequalities:

• \(2x + y \geq 10\) for protein
• \(x + 3y \geq 15\) for fat
• Non-negativity constraints \(x \geq 0\) and \(y \geq 0\)

These constraints form a feasible region that graphically represents all possible solutions. By analyzing this region, one can find the most efficient way to achieve objectives while satisfying all limits. Linear programming can be applied in many fields, from diet planning to logistics and finance, making it a versatile and powerful tool.

Graphing Inequalities

Graphing inequalities is a way to visually represent a system of inequalities. This helps in understanding the set of possible solutions that satisfy all conditions. In our cat food manufacturer example, the inequalities describe the amounts of fish and beef needed.
To graph these inequalities, you start by plotting each inequality on a coordinate plane. Draw the boundary line for each inequality by treating the inequality as an equation (e.g., for \(2x + y \geq 10\), draw the line \(2x + y = 10\)). Use a solid line if the inequality includes equal to or a dashed line if it does not.

• Determine which side of the line satisfies the inequality by picking a test point, usually the origin \((0, 0)\), unless it lies on the line.
• If the test point satisfies the inequality, shade that side of the line.
• Repeat this for each inequality.

The overlap of shaded areas from all inequalities gives the solution set. This visual guide is crucial in finding feasible solutions in linear programming.

Mathematical Modeling

Mathematical modeling involves creating a mathematical representation of a real-world situation. This process simplifies complex systems into models that can be analyzed and studied to predict outcomes and solve problems.
In our case of making cat food, mathematical modeling is used to determine how much fish and beef to combine to meet nutritional requirements. By defining variables \(x\) as ounces of fish and \(y\) as ounces of beef, and using the given nutrient data, we can model the problem with inequalities:

• \(12x + 6y \geq 60\) for protein
• \(3x + 9y \geq 45\) for fat

Through mathematical modeling, real-life issues are translated into mathematical language, enabling us to apply analytical techniques like linear programming. These models are invaluable not only in manufacturing but also in different sectors where resource optimization is crucial, such as economics and engineering.