Problem 24 - Algebra and Trigonometry with Analytic Geometry

Question

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 eat 16 kg of leaves and 17 kg of aquatic plants for maximum energy.

Step 1: Define the Variables

Let \( x \) be the kilograms of tree leaves and \( y \) be the kilograms of aquatic plants consumed daily by the moose.

Step 2: Formulate the Constraints

We have two main constraints:1. The total weight of food should not exceed 33 kg: \( x + y \leq 33 \).2. The moose needs at least 17 kg of aquatic plants: \( y \geq 17 \).Additionally, since negative consumption is not possible, \( x \geq 0 \) and \( y \geq 0 \).

Step 3: Establish the Objective Function

The objective is to maximize daily energy intake. Given that leaves provide four times the energy of aquatic plants, let the energy provided by a kilogram of aquatic plants be \( E \). Thus, leaves provide \( 4E \) energy per kilogram. The total energy function is then given by: \( z = 4E \cdot x + E \cdot y \).

Step 4: Simplify the Objective Function

Simplify the energy function by factoring out \( E \): \[ z = E (4x + y) \]To maximize energy, we need to maximize \( 4x + y \).

Step 5: Solve the Linear Programming Problem

We need to maximize \( 4x + y \) subject to:- \( x + y \leq 33 \)- \( y \geq 17 \)- \( x \geq 0 \)- \( y \geq 0 \).Let's graph these constraints:- The line \( x + y = 33 \) intersects the axes at \((33,0)\) and \((0,33)\).- The line \( y = 17 \) is horizontal.- Valid region is bounded by \( y = 17 \) and \( x + y = 33 \) in Quadrant I.

Step 6: Find the Vertices of the Feasible Region

The vertices of the feasible region are points of intersection of the lines and the axes:1. Intersection of \( y = 17 \) and \( x + y = 33 \) gives \( x = 16 \), so point is \((16, 17)\).2. Intersection with the y-axis at \((0, 17)\).3. Intersection of the line \( x + y = 33 \) with the x-axis provides point \((33, 0)\).

Step 7: Evaluate the Objective Function at Each Vertex

Compute \( 4x + y \) at each vertex:- At \((16, 17)\): \( 4(16) + 17 = 64 + 17 = 81 \).- At \((0, 17)\): \( 4(0) + 17 = 17 \).- At \((33, 0)\): \( 4(33) + 0 = 132 \).

Step 8: Verify and Conclude the Solution

The maximum energy intake is reached at \((33, 0)\), with energy value 132. However, \( y \geq 17 \) must be satisfied, so feasible maximum occurs at point \((16, 17)\) yielding energy 81.

Key Concepts

Optimization

In the context of linear programming, optimization refers to the process of finding the most efficient or effective solution to a given problem from a set of possible solutions. Here, our task is to optimize the blend of tree leaves and aquatic plants to maximize the moose's daily energy intake. To achieve this, we aim to select the best combination of these two types of food given specific guidelines and restrictions. The essence of optimization lies in its objective: making the best possible decision given certain constraints. For the moose, we seek the highest energy intake while considering consumption limits. This involves analyzing the relationship between different variables and selecting the values that yield the greatest result. This process represents the core objective in many real-world problems such as cost minimization, resource allocation, and profit maximization.

Constraints

Constraints in linear programming define the boundaries within which we have to operate. In our problem, constraints ensure that the moose's dietary needs and food limits are respected. They restrict the conditions under which food consumption is allowed, thus shaping the entire optimization process. Key constraints for our moose include:

• A total daily consumption cap of 33 kg of food – representing a hard limit based on dietary capacity: \( x + y \leq 33 \).
• A minimum intake requirement of 17 kg of aquatic plants to meet sodium levels: \( y \geq 17 \).
• Non-negativity constraints ensuring the moose can only consume non-negative quantities of leaves and plants: \( x \geq 0 \) and \( y \geq 0 \).

The constraints guide the linear programming model by narrowing down the area where the solution is searched, making them integral to the optimization process.

Objective Function

An objective function in linear programming is the formula that needs to be maximized or minimized. In this exercise, it revolves around maximizing the moose's energy intake from its diet. The objective function transforms the real-world aim into a mathematical representation that can be optimized.For the moose's diet:

• Let the energy from one kilogram of aquatic plants be \( E \).
• Leaves, providing four times this amount, contribute \( 4E \) energy per kilogram.

Thus, the energy objective function is expressed as \( z = 4E \cdot x + E \cdot y \). By factoring out \( E \), the task becomes maximizing \( 4x + y \). This equation is key, as it allows evaluation of different combinations of \( x \) and \( y \) to find the optimal, energy-maximizing configuration.

Feasible Region

The feasible region is the space defined by all potential combinations of variables that meet the linear programming model's constraints. In graphical terms, it is the area on a graph where all constraints overlap, providing a "solution space" from which the optimal point is selected.For our moose diet scenario, the feasible region is bounded by:

• The line \( x + y = 33 \), restricting total food intake.
• The line \( y = 17 \), enforcing the minimum aquatic plant consumption.
• The axes, where \( x \geq 0 \) and \( y \geq 0 \).

Within this region, points are evaluated to identify vertices—corners of the polygonal area—where potential optimal solutions may lie. Each vertex represents a specific combination of food and must be checked against the objective function. Only points within this feasible region are considered realistic options, ensuring all dietary restrictions are satisfied.