Problem 54 - Precalculus Mathematics for Calculus

Question

A cat food manufacturer uses fish and beef byproducts. The fish contains \(12 \mathrm{g}\) of protein and \(3 \mathrm{g}\) of fat per ounce. The beef contains \(6 \mathrm{g}\) of protein and \(9 \mathrm{g}\) 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

Use the system: \( 12x + 6y \geq 60 \), \( 3x + 9y \geq 45 \), \( x \geq 0 \), \( y \geq 0 \). Graph the solution region.

Step 1: Understanding the Problem

We need to create a system of inequalities that reflect the nutritional requirements for a can of cat food. The requirements are for at least 60 grams of protein and 45 grams of fat per can, using fish and beef byproducts.

Step 2: Variables Definition

Define variables: Let \( x \) be the ounces of fish, and \( y \) be the ounces of beef to be used in the cat food.

Step 3: Create Inequality for Protein

Calculate the protein contributed by both fish and beef. Fish provides 12 grams and beef provides 6 grams of protein per ounce. Thus, the inequality becomes \( 12x + 6y \geq 60 \).

Step 4: Create Inequality for Fat

Calculate the fat contributed by both fish and beef. Fish provides 3 grams and beef provides 9 grams of fat per ounce. Thus, the inequality becomes \( 3x + 9y \geq 45 \).

Step 5: Non-negativity Constraints

The amounts of fish and beef used cannot be negative, so we have \( x \geq 0 \) and \( y \geq 0 \).

Step 6: Graph the Inequalities

Plot the lines of the inequalities \( 12x + 6y = 60 \) and \( 3x + 9y = 45 \). "Fill" the region where both inequalities and the non-negativity constraints are satisfied. This region is the feasible region.

Step 7: Identify the Solution Set

The solution set is the area of overlap on the graph where all the inequalities are satisfied simultaneously. It represents the set of possible values of ounces of fish and beef to meet the nutritional requirements.

Key Concepts

System of Inequalities

In every system of inequalities, we aim to find a set of inequalities that portray a specific set of rules or conditions that need to be met. In this case, we are dealing with creating a system involving nutritional requirements for cat food.
The problem involves ensuring that the product contains enough protein and fat. We achieve this by setting up at least two inequalities:

• One inequality to account for protein requirements. Because fish provides 12g of protein per ounce and beef provides 6g, we express this as: \(12x + 6y \geq 60\).
• The second inequality is for fat. Since each ounce of fish provides 3g and beef provides 9g of fat, the inequality becomes: \(3x + 9y \geq 45\).
• We also need to include non-negativity constraints because negative amounts of ingredients aren't possible (i.e., \(x \geq 0\) and \(y \geq 0\)).

When these inequalities are combined, they form a system of inequalities. This system tells us where the conditions for both protein and fat overlap and meet the specified requirements.

Linear Programming

Linear programming is a mathematical method used to determine the best possible outcome, such as maximum profit or lowest cost, in a given model. This model typically involves a linear objective function and a set of linear inequalities, or constraints.
In the context of this exercise, we are not necessarily trying to optimize, but rather identify the feasible set - the values that meet all nutritional constraints.

• The objective function in traditional linear programming models could be, for instance, a goal to minimize cost or maximize nutrition within budget constraints.
• Here, however, we just aim to ensure that the system of constraints is satisfied.
• This often involves graphing, especially in two-variable cases like ours, to visually find solutions within the feasible region.

Thus, while linear programming often involves finding the optimal solution in constrained environments, sometimes simply determining a suitable strategy that works for all given constraints is its application, just as in managing nutritional content in cat food.

Graphing Inequalities

Graphing inequalities involves plotting lines or curves that represent boundaries defined by an inequality, and shading the region that satisfies the inequality.
The process essentially converts algebraic conditions into geometric areas, making it easier to visualize potential solutions.
For the problem at hand, the process involves:

• Plotting the lines represented by the equations \(12x + 6y = 60\) for protein and \(3x + 9y = 45\) for fat on a coordinate plane.
• Notice where these lines intersect and the shaded areas which represent solution sets. These areas show all combinations of fish (\(x\)) and beef (\(y\)) that meet or exceed the required protein and fat values.
• The feasible region is typically the area where all conditions match, including the non-negative constraints \(x \geq 0\) and \(y \geq 0\).

Graphing allows us to visually inspect the feasible region, which represents all the possible values that meet all constraints together. It's a powerful tool for understanding how different inequality constraints interact and overlap.