Problem 24 - Precalculus: Functions and Graphs

Question

A moose's diet A moose feeding primarily on tree leaves and aquatic plants is capable of digesting no more than 33 kilograms of these foods daily. Although the aquatic plants are lower in energy content, the animal must eat at least 17 kilograms to satisfy its sodium requirement. A kilogram of leaves provides four times as much energy as a kilogram of aquatic plants. Find the combination of foods that maximizes the daily energy intake.

Step-by-Step Solution

Answer

The moose should consume 16 kg of leaves and 17 kg of aquatic plants.

Step 1: Define the Variables

Let \( x \) represent the kilograms of tree leaves consumed by the moose and \( y \) represent the kilograms of aquatic plants consumed. The goal is to maximize the energy intake.

Step 2: Set Up the Equations

Since a kilogram of leaves provides four times as much energy as a kilogram of aquatic plants, we express the total energy intake as \( E = 4x + y \). This is the objective function to be maximized.

Step 3: Establish the Constraints

The two main constraints in the problem are: 1. Total weight: \( x + y \leq 33 \).2. Sodium requirement: \( y \geq 17 \).These constraints must be met for the solution to be valid.

Step 4: Apply Linear Programming Feasibility

Graph the constraints on a coordinate plane, with \( x \) on the horizontal axis and \( y \) on the vertical axis. The feasible region is the area where the inequalities \( x + y \leq 33 \) and \( y \geq 17 \) intersect.

Step 5: Identify the Critical Points

The vertices (corners) of the feasible region are calculated by solving the system of equations formed by the intersecting lines: 1. \( x + y = 33 \and y = 17 \) gives \((16, 17)\).2. On the y-axis: \( y = 17 \), thus \( x = 0, y = 17 \).3. On the constraint: \( x = 0 \and x + y = 33 \), thus \((0, 33)\).

Step 6: Calculate Energy at Each Vertex

Substitute the vertex coordinates into the objective function \( E = 4x + y \):1. At \((16, 17)\): \( E = 4(16) + 17 = 64 + 17 = 81 \).2. At \((0, 17)\): \( E = 4(0) + 17 = 17 \).3. At \((0, 33)\): \( E = 4(0) + 33 = 33 \).

Step 7: Choose the Maximum Energy Point

Comparing the calculated energies, the maximum energy of 81 is obtained at the vertex \((16, 17)\). This means the moose should consume 16 kg of leaves and 17 kg of aquatic plants for the highest energy intake.

Key Concepts

Energy Maximization

In linear programming, energy maximization is about finding the best combination of resources. These resources provide the most value or benefit to a system—in this case, a moose's diet. For the moose, maximizing energy means choosing the right mix of tree leaves and aquatic plants. Each type contributes differently to the total energy intake.

• Leaves contribute significantly more energy.
• Aquatic plants, although necessary, have lower energy content.

This scenario is framed by an objective function: \(E = 4x + y\). Here, \(x\) and \(y\) represent the kilograms of leaves and aquatic plants, respectively. The coefficient '4' signifies that leaves provide four times more energy than the aquatic plants.To solve such a problem, one must determine the right balance where energy intake reaches its peak, within given limits. These constraints ensure that the solution remains practical and realistic.

Constraint Satisfaction

Constraint satisfaction in linear programming involves solving problems while adhering to set limits. These limits are expressed as inequalities that must be satisfied for a solution to be viable. For the moose, the following constraints are vital:

• The total consumption should not exceed 33 kg, i.e., \(x + y \leq 33\).
• A minimum of 17 kg of aquatic plants is needed, i.e., \(y \geq 17\).

These constraints reflect the natural limitations and nutritional needs of the moose. They ensure that the solution not only maximizes energy but also respects these physical and dietary boundaries.In linear programming, satisfying these constraints while solving the optimization problem is crucial. It means that the solution is both optimal and feasible in real-world terms.

Feasible Region

The feasible region in a linear programming problem is the area on a graph where all constraints overlap. This region represents all possible solutions that meet the given constraints.For the moose's diet, two constraints create a specific feasible region on the graph:

• The maximum combined weight is 33 kg, represented by the line \(x + y = 33\).
• A minimum of 17 kg of aquatic plants, indicated by the line \(y = 17\).

By plotting these constraints, we find the area where solutions are possible. The feasible region is where the constraints intersect. In our case, this forms a polygon on the graph. Within this region, we look for the maximum energy point.Identifying and evaluating the vertices (corners) of this polygon helps determine the best solution. This process ensures that the moose's diet not only maximizes energy but also follows the dietary rules established.