Problem 17 - Algebra and Trigonometry with Analytic Geometry

Question

Three substances, X, Y, and Z, each contain four ingredients, A, B, C, and D. The percentage of each ingredient and the cost in cents per ounce of each substance are given in the following table. $$ \begin{array}{|c|cccc|c|} \hline & \multicolumn{4}{|c|}{\text { Ingredients }} & \text { Cost per } \\ \cline { 2 - 5 } \text { Substance } & \mathbf{A} & \mathbf{B} & \mathbf{C} & \mathbf{D} & \text { ounce } \\ \hline \mathrm{X} & 20 \% & 10 \% & 25 \% & 45 \% & 25 \% \\ \mathrm{Y} & 20 \% & 40 \% & 15 \% & 25 \% & 35 \% \\ \mathrm{Z} & 10 \% & 20 \% & 25 \% & 45 \% & 50 \% \\ \hline \end{array} $$ If the cost is to be minimal, how many ounces of each substance should be combined to obtain a mixture of 20 ounces containing at least \(14 \% \mathrm{~A}, 16 \% \mathrm{~B}\), and \(20 \% \mathrm{C}\) ? What combination would make the cost greatest?

Step-by-Step Solution

Answer

Use linear programming to find the combination that minimizes or maximizes the cost, meeting the ingredient criteria.

Step 1: Define the Variables

Let the variables \(x\), \(y\), and \(z\) represent the ounces of substances X, Y, and Z, respectively, in the mixture. We need to find the values of \(x\), \(y\), and \(z\) that satisfy the ingredient requirements and make the cost either minimal or maximal.

Step 2: Set Up the Ingredient Constraints

Based on the percentage constraints for each ingredient: - For ingredient A: \(0.20x + 0.20y + 0.10z \geq 0.14 \, \times \, 20\).- For ingredient B: \(0.10x + 0.40y + 0.20z \geq 0.16 \, \times \, 20\).- For ingredient C: \(0.25x + 0.15y + 0.25z \geq 0.20 \, \times \, 20\).These must be satisfied with \(x + y + z = 20\).

Step 3: Convert Percentage to Ounce Constraints

Calculate the required ounces for each ingredient:- For A: \(0.20x + 0.20y + 0.10z \geq 2.8\).- For B: \(0.10x + 0.40y + 0.20z \geq 3.2\).- For C: \(0.25x + 0.15y + 0.25z \geq 4\).

Step 4: Cost Function

The cost function to minimize is: \[ 25x + 35y + 50z \]This represents the total cost in cents based on the given cost per ounce for each substance.

Step 5: Solve for Minimum Cost

Using linear programming or substitution and trial method, solve the set of inequalities including the primary equation \(x+y+z=20\) and find the values of \(x\), \(y\), and \(z\) that minimize the cost. Typically, computational software or systematic substitution would be used.

Step 6: Solve for Maximal Cost

Similarly, substitute and solve the inequalities for conditions when the cost function: \[ 25x + 35y + 50z \]is maximal while still satisfying the constraints. This explores the opposite extremes of ingredient proportions within the bounds of the constraints.

Key Concepts

Optimization

Linear programming is a powerful mathematical technique used to find the optimal solution to a problem by maximizing or minimizing a particular function. In the case of our exercise, we need to find the right combination of substances X, Y, and Z to either minimize or maximize the cost.
To do this, linear programming employs a set of constraints (rules the solution must respect), and an objective function, which is the formula you want to optimize. In this exercise, our constraints are based on the required amounts of ingredients A, B, and C in the mixture. Meanwhile, the objective function is the total cost at which we want to aim—either minimize or maximize.
Solving such a linear optimization problem can sometimes be done by hand using techniques such as graphing or substitution, but it's more efficiently handled using computational tools, as these can quickly evaluate numerous possibilities within the constraints. These tools seek to find the point where the objective function reaches its lowest or highest value while maintaining all the given constraints intact.

Cost Minimization

Minimizing cost is a common goal in linear programming. In our exercise, we aim to blend substances X, Y, and Z in such a way that the total cost per ounce of the mixture is minimized. To achieve this, we need to carefully consider the cost per ounce of each substance and weigh it against how it contributes to meeting the ingredient requirements.
The cost function to be minimized is expressed as:

• \( 25x + 35y + 50z \)

This mathematical expression encapsulates the total cost of the mixture based on the costs of substances X, Y, and Z, where:

• \( x \) - ounces of substance X at 25 cents each
• \( y \) - ounces of substance Y at 35 cents each
• \( z \) - ounces of substance Z at 50 cents each

The goal is to minimize this equation while satisfying all the ingredient constraints. This can involve juggling different ingredient proportions to find the point where all constraints intersect at the lowest possible cost. Often, the optimal solution will lie on the boundary of the feasible region formed by these constraints.

Ingredient Constraint

Ingredient constraints define the minimum percentage of each ingredient required in the mixture. These constraints ensure that the final product maintains a certain standard or quality by having essential ingredients in the specified quantities.
In the context of our problem, we have constraints based on ingredients A, B, and C:

• For Ingredient A: \(0.20x + 0.20y + 0.10z \geq 2.8\)
• For Ingredient B: \(0.10x + 0.40y + 0.20z \geq 3.2\)
• For Ingredient C: \(0.25x + 0.15y + 0.25z \geq 4\)

These constraints are derived from the percentage requirements transformed into actual ounce amounts, reflecting the composition needed for 20 ounces of the mixture.
To comply, our solution must meet or exceed these amounts, which signifies that any combination of X, Y, and Z must satisfy these inequalities to be deemed valid. Understanding and managing these constraints is a critical part of using linear programming to find a feasible and optimal solution.