Problem 93 - A Graphical Approach to College Algebra

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 4000 medical kits and 2000 water containers to aid 36,000 people.

Step 1: Understanding the Problem

We need to send medical kits and water containers to maximize aid to earthquake victims. Each medical kit aids 4 people and has a weight of 10 pounds and a volume of 1 cubic foot, while each water container aids 10 people and weighs 20 pounds and has a volume of 1 cubic foot. The plane can carry up to 80,000 pounds and 6000 cubic feet.

Step 2: Define Variables

Let \( x \) represent the number of medical kits and \( y \) represent the number of water containers. We need to determine the values of \( x \) and \( y \) that maximize the total number of people aided, subject to weight and volume constraints.

Step 3: Formulate Objective Function

Our goal is to maximize the number of people aided, which is expressed by the objective function: \( Z = 4x + 10y \), representing the total number of people aided by the medical kits and water containers.

Step 4: Set Up Constraints

The constraints are based on the weight and volume limits of the plane:1. Weight Constraint: \(10x + 20y \leq 80,000\)2. Volume Constraint: \(x + y \leq 6000\) 3. Non-negativity: \(x, y \geq 0\)

Step 5: Solve the System of Constraints

We need to graph the inequalities and find the feasible region that satisfies all constraints. The vertex points of this region will be considered to maximize the objective function. We focus calculations on the vertices:- Intersection of \(10x + 20y = 80,000\) and \(x + y = 6000\).

Step 6: Calculate Vertices

Solve for intersections:1. Solve \(10x + 20y = 80,000\) and \(x + y = 6000\) simultaneously: - \( x + y = 6000 \Rightarrow y = 6000 - x \) - Substitute \( y \) in \( 10x + 20y = 80,000 \): \[ 10x + 20(6000 - x) = 80,000 \] \[ 10x + 120,000 - 20x = 80,000 \] \[ -10x = -40,000 \Rightarrow x = 4,000 \] \[ y = 6000 - 4000 = 2000 \]

Step 7: Evaluate Objective Function at Vertices

At the vertex \((x, y) = (4000, 2000)\), calculate the objective function: \[ Z = 4(4000) + 10(2000) = 16,000 + 20,000 = 36,000 \] people. Thus, the maximum possible number of people aided is \(36,000\).

Step 8: Final Solution

Send 4000 medical kits and 2000 water containers to maximize the number of people aided, which will aid 36,000 people.

Key Concepts

Objective Function

In linear programming problems like this, the objective function is a crucial component. Here, it's used to represent the main goal of the problem, which is to maximize the number of people aided through the shipment of medical kits and water containers.
We express this goal mathematically with an equation: \[Z = 4x + 10y\]where:

• \( x \) is the number of medical kits, each aiding 4 people, translating to \(4x\).
• \( y \) is the number of water containers, each aiding 10 people, translating to \(10y\).

The larger value of \(Z\), the more people are helped. Our task is to find the values of \(x\) and \(y\) that will maximize \(Z\), given the constraints of weight and volume of the cargo plane.

Constraints

Constraints in linear programming are the rules or conditions that the solution must satisfy. In this problem, the constraints are derived from the plane's weight and volume limitations.
The constraints are formulated as:

• **Weight Constraint:** The combined weight of the kits and containers cannot exceed 80,000 pounds:\[10x + 20y \leq 80,000\]
• **Volume Constraint:** The total volume of the kits and containers must be less than or equal to 6000 cubic feet:\[x + y \leq 6000\]
• **Non-negativity Constraints:** You cannot ship a negative number of items:\[x \geq 0, \, y \geq 0\]

These constraints are essential to ensuring the solution is feasible under real-world conditions. By graphing these inequalities, we can identify the feasible region where these constraints are satisfied.

Feasible Region

The feasible region in linear programming represents the set of all points that satisfy all the constraints. In this problem, it is the area within which we can operate while staying within the weight and volume limits of the plane.
To find the feasible region, we graph our constraints on a coordinate system where \( x \) and \( y \) are the axes:- The line \(10x + 20y = 80,000\) represents the maximum weight.- The line \(x + y = 6000\) represents the maximum volume.- Both lines are bounded by the axes, considering the non-negativity constraints.
The area where all of these constraints overlap is the feasible region. Solutions within this area are potential solutions that can be tested to see which one maximizes the objective function.

Vertices Evaluation

Vertices evaluation is a method used in linear programming to find the optimal solution by testing the objective function at each corner (vertex) of the feasible region.
For this exercise, we focus on the intersections of the constraints, as these corners or vertices are where the extreme values of the objective function are likely to be found.
We evaluate the intersection point of the constraints \(10x + 20y = 80,000\) and \(x + y = 6000\). Solving these equations gives us the vertex:

• \((x, y) = (4000, 2000)\)

Here we calculate the objective function:\[Z = 4(4000) + 10(2000) = 36,000\]This result indicates that at the vertex \((4000, 2000)\), the solution maximizes the number of people aided to 36,000, making it the best outcome under the given conditions.