Problem 109 - A Graphical Approach to Precalculus with Limits

Question

Earthquake victims need medical supplies and bottled water. Each medical kit measures 1 cubic foot and weighs 10 pounds. Each container of water is also 1 cubic foot, but weighs 20 pounds. The plane can carry only \(80,000\) pounds, with total volume 6000 cubic feet. Each medical kit will aid 4 people, while each container of water will serve 10 people. How many of each should be sent in order to maximize the number of people aided? How many people will be aided?

Step-by-Step Solution

Answer

Send 4,000 medical kits and 2,000 water containers to aid 36,000 people.

Step 1: Define the Variables

Let's denote \(x\) as the number of medical kits and \(y\) as the number of water containers. This helps to set up the problem in terms of mathematical variables.

Step 2: Write the Constraints

We have two constraints based on the weight and volume:1. Weight constraint: \(10x + 20y \leq 80,000\)2. Volume constraint: \(x + y \leq 6,000\)These inequalities ensure that both weight and volume capacities of the plane are not exceeded.

Step 3: Formulate the Objective Function

We aim to maximize the number of people aided. Therefore, the objective function is:\[ P = 4x + 10y \]where \(P\) is the total number of people aided by the medical kits and water containers.

Step 4: Solve the System of Inequalities

First, simplify and solve the constraint inequalities:1. Divide the weight constraint by 10: \(x + 2y \leq 8,000\) Now, find points of intersection and region of feasibility in the \(xy\)-plane for all constraints:- Intersection of \(x + y = 6,000\) and \(x + 2y = 8,000\) by solving: \[ \begin{cases} x + y = 6,000 \ x + 2y = 8,000 \end{cases} \] Solving these gives: \(y = 2,000\), \(x = 4,000\).- Check boundary points like \((x=0, y=3,000), (x=6,000, y=0)\) inside inequalities as potential solutions.

Step 5: Evaluate Objective Function at Feasible Points

Evaluate \(P = 4x + 10y\) at key feasible boundary points:- At \((0, 3,000)\), \(P = 4(0) + 10(3,000) = 30,000\)- At \((6,000, 0)\), \(P = 4(6,000) + 10(0) = 24,000\)- At \((4,000, 2,000)\), \(P = 4(4,000) + 10(2,000) = 36,000\)The maximum number of people aided is at \((4,000, 2,000)\) with \(P = 36,000\).

Step 6: Analyze the Best Choice for Aid

The best choice is to send 4,000 medical kits and 2,000 water containers. This configuration uses the entire weight and volume capacity available on the plane most efficiently.

Key Concepts

Objective Function

In linear programming, the Objective Function is the expression that needs to be maximized or minimized. It's essentially the end goal of the problem-solving process. In our exercise, the objective is to maximize the number of people aided by maximizing the combination of medical kits and water containers.
The Objective Function is formulated by expressing the total 'aid' received from combining medical kits and water containers. Since each medical kit aids 4 people and each container of water aids 10, the function is structured as follows:

• Define the total aid: \( P = 4x + 10y \)

Here, \( x \) is the number of medical kits, and \( y \) is the number of water containers.
Solving the objective function means finding the optimal numbers of \( x \) and \( y \) that yield the highest possible value of \( P \), given the constraints. This ensures we benefit the maximum number of people possible within the specified limits of the plane's carrying capacity.

Constraints

Constraints in linear programming define the limitations or conditions that the variables must satisfy. They are typically expressed in the form of inequalities.
In the given exercise, there are two key constraints—weight and volume:

• Weight constraint: Every medical kit weighs 10 pounds, and each container of water weighs 20 pounds. The total weight capacity is 80,000 pounds, leading to the inequality \( 10x + 20y \leq 80,000 \).
• Volume constraint: Both the medical kit and the water container occupy 1 cubic foot each, with a combined total volume restriction of 6,000 cubic feet. This gives us \( x + y \leq 6,000 \).

These constraints must be satisfied to ensure the safe transport of supplies without exceeding the plane's limits. They play a vital role in determining the feasible solutions for the objective function.

Feasibility Region

The Feasibility Region is the set of all possible solutions that meet the constraints. This region is graphically represented in the \( xy \)-plane where all the inequalities intersect.
In solving the exercise, the constraints \( 10x + 20y \leq 80,000 \) and \( x + y \leq 6,000 \) create a polygonal region on the graph. This is the feasible region where any point within or on the boundary can be a potential solution.
The vertices of this region are especially important because, in linear programming, the optimal solution often lies at these corners. Evaluating the objective function at these key points helps to find the maximum or minimum value efficiently. In this problem, the point \((4,000, 2,000)\) provided the maximum number of people aided.

System of Inequalities

A System of Inequalities consists of two or more linear inequalities that collectively define the constraints in linear programming. They guide the search for optimal solutions within a feasible range.
The system in the exercise included:

• \( 10x + 20y \leq 80,000 \) (weight)
• \( x + y \leq 6,000 \) (volume)

By considering all inequalities together, we ensure that both constraints are met simultaneously.
To solve, we simplify these inequalities where possible—like dividing by constants to make them easier to graph and analyze. Then, by solving the system, such as finding intersection points \((x, y)\), we can determine the vertices of the feasibility region. These steps are essential to identify solutions that not only satisfy all constraints but also optimize the objective function for the best outcome.