Problem 11 - Algebra and Trigonometry with Analytic Geometry

Question

Two substances, \(S\) and T, each contain two types of ingredients, I and \(G\). One pound of \(S\) contains 2 ounces of I and 4 ounces of \(G\). One pound of \(T\) contains 2 ounces of I and 6 ounces of G. A manufacturer plans to combine quantities of the two substances to obtain a mixture that contains at least 9 ounces of \(\mathrm{I}\) and 20 ounces of \(\mathrm{G}\). If the cost of \(S\) is $$\$ 3$$ per pound and the cost of \(T\) is $$\$ 4$$ per pound, how much of each substance should be used to keep the cost to a minimum?

Step-by-Step Solution

Answer

Use 2 pounds of \( S \) and 2.5 pounds of \( T \).

Step 1: Define the Variables

Let \( x \) be the number of pounds of substance \( S \), and \( y \) be the number of pounds of substance \( T \). These are the quantities we need to determine to meet the ingredient requirements at the lowest cost.

Step 2: Set Up the Inequalities for Ingredients

We need at least 9 ounces of \( I \) and 20 ounces of \( G \). Each pound of \( S \) contributes 2 ounces of \( I \) and each pound of \( T \) also contributes 2 ounces of \( I \). Therefore, we need at least \( 2x + 2y \geq 9 \). Each pound of \( S \) contributes 4 ounces of \( G \) and each pound of \( T \) contributes 6 ounces of \( G \), thus \( 4x + 6y \geq 20 \).

Step 3: Formulate the Cost Function

The cost function we want to minimize is \( C = 3x + 4y \), where \( x \) is the cost per pound of \( S \) and \( y \) is the cost per pound of \( T \).

Step 4: Solve the System of Inequalities

First solve the inequalities: 1. \( 2x + 2y \geq 9 \)2. \( 4x + 6y \geq 20 \)Solve the first equation for \( y \):\[ y \geq \frac{9 - 2x}{2} \].Solve the second equation for \( y \):\[ y \geq \frac{20 - 4x}{6} \].Graph these inequalities on the coordinate plane to find the feasible region.

Step 5: Find the Vertices of the Feasible Region

Graph the constraints and find where they intersect with each other or with the axes:- For \( 2x + 2y = 9 \): intersects the axes at \((4.5, 0)\) and \((0, 4.5)\).- For \( 4x + 6y = 20 \): intersects the axes at \((5, 0)\) and \((0, rac{10}{3})\).Check the intersection point of these lines by solving the system of linear equations. Set both expressions for \( y \) equal to each other and solve for \( x \):\[ \frac{9 - 2x}{2} = \frac{20 - 4x}{6} \]. Solving this gives the intersection \((2, 2.5)\).

Key Concepts

System of Inequalities

A "System of Inequalities" is a set of two or more inequalities with the same variables. In our problem, the manufacturer has to ensure that the quantities of ingredients I and G in the mixture satisfy certain minimum requirements. This creates a system of inequalities that must be resolved for effective cost minimization.

Let's consider two inequalities from the problem:

• The first inequality ensures that the amount of I is at least 9 ounces. Each pound of substance S and T provides 2 ounces of I, hence the inequality: \[ 2x + 2y \geq 9 \]
• The second inequality ensures that the amount of G is at least 20 ounces, with S contributing 4 ounces per pound and T contributing 6 ounces, leading to:\[ 4x + 6y \geq 20 \]

Solving these inequalities provides the set of all possible combinations of pounds of substances S and T that meet the requirements. This provides the feasible region which will be analyzed further in conjunction with the cost function.

Cost Minimization

"Cost Minimization" involves finding the lowest cost to meet the constraints set by the system of inequalities. The cost function in this problem is represented by the equation:\[ C = 3x + 4y \]

This function represents the total cost depending on the amounts of substances S and T used. Each pound of S costs \(3 and each pound of T costs \)4. The goal is to minimize this cost while still complying with the system of inequalities.

To achieve this, it is essential to determine the feasible region where all constraints are satisfied. The solution to the linear programming problem usually occurs at the vertices of this feasible region. By evaluating the cost at each vertex, you can find the combination of S and T that delivers the lowest cost, fulfilling all necessary constraints.

Graphing Inequalities

Graphing inequalities involves plotting each inequality on a coordinate plane to visualize the possible solutions. Each inequality defines a half-plane on the graph. The region where these half-planes overlap represents all solutions that satisfy all inequalities simultaneously.

For example, in the exercise:

• The inequality \( 2x + 2y \geq 9 \) creates a line dividing the plane. The region above this line satisfies the inequality.
• The inequality \( 4x + 6y \geq 20 \) similarly divides the plane, with the region above it containing all solutions for the inequality.

By plotting these lines, the intersection or overlap of regions denotes the feasible region. This is the practical set of solutions where both ingredient requirements for I and G are met. Calculating the points where these lines intersect with each other and the axes helps to accurately define this area. Once this graph is complete, identifying the vertices of this region will allow us to determine the optimal mix of S and T by applying the cost function.