Problem 74 - Precalculus: Mathematics for Calculus

Question

A cat food manufacturer uses fish and beef byproducts. 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 g of fat. Find a system of inequalities that describes the possible number of ounces of fish and beef byproducts that can be used in each can to satisfy these minimum requirements. Graph the solution set.

Step-by-Step Solution

Answer

The system is: \(12x + 6y \geq 60\), \(3x + 9y \geq 45\), \(x\geq0\), and \(y\geq0\).

Step 1: Define the Variables

Let's define the variables needed to form our system of inequalities. Let \( x \) represent the number of ounces of fish byproduct, and \( y \) represent the number of ounces of beef byproduct used.

Step 2: Write the Inequality for Protein

For protein, the requirement is at least 60 grams. Fish byproduct contributes 12 grams per ounce, and beef byproduct contributes 6 grams per ounce. Thus, the inequality for protein is: \[12x + 6y \geq 60\]

Step 3: Write the Inequality for Fat

For fat, the requirement is at least 45 grams. The fish byproduct provides 3 grams per ounce, and the beef byproduct provides 9 grams per ounce. Hence, the inequality for fat is: \[3x + 9y \geq 45\]

Step 4: Non-Negative Constraints

Since you cannot use a negative amount of fish or beef byproduct, you must have the conditions: \[x \geq 0 \] and \[y \geq 0 \]

Step 5: Graph the Solution Set

Plot the inequalities on a coordinate plane with \(x\) and \(y\) axes. The feasible region will be where all inequalities overlap, above or on the line defined by \(12x + 6y = 60\), above or on the line defined by \(3x + 9y = 45\), and in the first quadrant where \(x \geq 0\) and \(y \geq 0\).

Key Concepts

System of Inequalities

When tackling a problem like the one with the cat food manufacturer, we begin by framing it as a system of inequalities. This framework helps us express constraints on the quantities of resources to be used—in this case, fish and beef byproducts.
The first step involves defining variables:

• Let variable \( x \) represent the number of ounces of fish byproduct.
• Let variable \( y \) denote the ounces of beef byproduct.

With these variables, we formulate inequalities based on the nutritional requirements:

• For protein, where the fish provides 12 grams per ounce and beef 6 grams per ounce, the inequality \(12x + 6y \geq 60\) ensures the minimum total protein is met.
• For fat, fish provides 3 grams per ounce and beef 9 grams per ounce, forming the inequality \(3x + 9y \geq 45\) to cover the required fat amount.

Non-negativity constraints \(x \geq 0\) and \(y \geq 0\) ensure negative values do not appear in the solution. Together, these inequalities create a structured way to visualize and solve the manufacturer's problem.

Graphing Inequalities

Graphing inequalities is a powerful technique that helps visualize solutions to linear programming problems. For our exercise, graphing is the step where mathematical concepts transition into a geometric interpretation.
Begin by drawing the coordinate axes where the x-axis represents the ounces of fish byproduct (\(x\)), and the y-axis represents the ounces of beef byproduct (\(y\)). Each inequality can be graphed as a straight line:

• For \(12x + 6y \geq 60\), divide the inequality to transform it into a line equation: \(2x + y = 10\).
• For \(3x + 9y \geq 45\), simplify to \(x + 3y = 15\).

Each line divides the plane into two regions. The region containing points where the inequality holds is the one to focus on. Shade this area on the plane to represent all possible solutions meeting that specific constraint.

Feasible Region

The feasible region is the heart of solving linear programming problems, representing all possible solutions that simultaneously satisfy all the system's inequalities. For the cat food problem, this is the overlap of shaded regions defined by our inequalities.
When graphing maybe challenging, here’s what to look for:

• Ensure the shaded area above or on the line \(12x + 6y = 60\) remains, as this represents all combinations where enough protein is included.
• Similarly, check that the region above or on \(3x + 9y = 45\) line is maintained, signifying the minimum fat requirement is met.
• The first quadrant is always part of the permissible region since using negative quantities of byproducts isn't feasible.

The feasible region, therefore, is a bounded area on the plane where these inequality conditions overlap, adhering to the nutritional requirements laid out by producers.