Problem 106 - A Graphical Approach to Precalculus with Limits

Question

Theo requires two food supplements:I and II. He can get these supplements from two different products \(A\) and \(B,\) as shown in the following table.$$\begin{array}{l|c|c}\hline \text { Supplement (grams/serving) } & \text { I } & \text { II } \\\\\hline \text { Product } A & 3 & 2 \\\\\text { Product } B & 2 & 4\end{array}$$.Theo's physician recommends at least 15 grams of each supplement in Theo's daily diet. If product \(A\) costs \(25 \phi\) per serving and product \(B\) costs \(40 \notin\) per serving, how can he satisfy his requirements most economically?

Step-by-Step Solution

Answer

Buy 5 servings of product A and 0 servings of product B for the minimum cost.

Step 1: Define Variables

Let the number of servings of product \( A \) be \( x \) and the number of servings of product \( B \) be \( y \). We need to find the values for \( x \) and \( y \) that minimize cost while meeting the nutritional requirements.

Step 2: Set Up Inequalities

From the nutritional requirements, we have two inequalities: \( 3x + 2y \geq 15 \) for supplement I, and \( 2x + 4y \geq 15 \) for supplement II.

Step 3: Cost Function

The cost function to minimize is given by \( C = 25x + 40y \), where 25 and 40 are the costs per serving of products \( A \) and \( B \) respectively.

Step 4: Graph Constraints

Graph the lines represented by the inequalities \( 3x + 2y = 15 \) and \( 2x + 4y = 15 \). The feasible region is bound by these lines on or above them, and the axes.

Step 5: Identify Feasible Solution

The feasible region determined by the intersection of \( 3x + 2y \geq 15 \), \( 2x + 4y \geq 15 \), \( x \geq 0 \), and \( y \geq 0 \) must be found next. Check the vertices of this region for optimization.

Step 6: Evaluate Vertices

Calculate the total cost \( C = 25x + 40y \) at each vertex of the feasible region. Common points to test arise where lines intersect each other or the axes, such as (0,7.5), (5,0), and (3,3).

Step 7: Choose the Minimum Cost

Compute the costs: \( C(0, 7.5) = 300 \), \( C(5, 0) = 125 \), \( C(3, 3) = 195 \). The minimum cost is at \( (5, 0) \) with \( C = 125 \phi\).

Key Concepts

Optimization

In the context of linear programming, optimization is all about finding the best solution to a problem. This involves selecting values that result in either the maximum or minimum outcome according to a defined criterion. In Theo's case, he wants to minimize the cost of purchasing food supplements while keeping his nutrient intake balanced. The goal of optimization here is to achieve an optimal balance by using specific amounts of Product A and Product B, ensuring he spends the least amount of money while still meeting his dietary needs. Optimization often requires setting up a system of equations or inequalities and evaluating different possible solutions to pinpoint the most cost-effective choice. This process entails understanding both the requirements Theo must meet and the costs associated with each choice, such as:

• Meeting at least 15 grams of each required supplement.
• Determining how many servings of each product to buy.
• Calculating the total cost to find the least expensive option.

Cost Function

The cost function is a mathematical expression that describes how Theo's total spending changes based on the quantities of products A and B that he buys. It's a crucial component of any optimization problem because it defines what needs to be minimized.For this scenario, the cost function is expressed as\[C = 25x + 40y\]where:

• \(x\) is the number of servings of Product A.
• \(y\) is the number of servings of Product B.
• 25 and 40 are the costs per serving of Products A and B, respectively.

By using the cost function, Theo can see how much he will need to spend depending on different combinations of \(x\) and \(y\). The function is indispensable in the search for the minimal cost subject to the dietary constraints. Finding the point where this cost is the lowest will result in the optimal buying strategy.

Inequality Constraints

Inequality constraints are expressions that limit the possible solutions in an optimization problem. They ensure that the solution respects all the given conditions, like the nutritional requirements Theo's doctor set for the food supplements.In our scenario, these inequality constraints are expressed as:

• \(3x + 2y \geq 15\): Ensures Theo gets at least 15 grams of Supplement I.
• \(2x + 4y \geq 15\): Ensures that same minimum for Supplement II.

These constraints form a feasible region on the coordinate plane where all viable solutions exist. They are not hard barriers but thresholds that the solution must meet or exceed. To ensure a successful minimization of Theo's cost, it's essential to respect these constraints, as any solution lying outside this region is not valid or feasible. Evaluating points on the boundary of this region helps in identifying not just any solution, but the most economic one.