In the realm of mathematical optimization, evaluating profit functions at vertices is a cornerstone technique, particularly in linear programming. Guys, if you're diving into this field, understanding how to maximize profit or minimize costs is crucial. This article will walk you through the process step by step, making it super easy to grasp. We'll break down the concept, show you the calculations, and explain why this method is so effective. Let's get started and unlock the secrets of profit optimization!
Understanding the Profit Function
Before we dive into the calculations, let's make sure we're all on the same page about what a profit function is. In simple terms, a profit function is a mathematical expression that shows how profit depends on different factors, like the quantity of products sold or resources used. In our case, the profit function is given by:
Here, P represents the total profit, x and y are variables (think of them as quantities of two different products), and the coefficients (0.04, 0.05, and 0.06) represent the profit per unit of each variable or a combination of them. The term 16 - x - y introduces a constraint, meaning that the sum of x, y, and this term cannot exceed 16. This is super common in real-world scenarios where resources are limited.
Now, why is this important? Well, in many business and operational scenarios, you want to figure out how to maximize your profit, right? This is where the concept of vertices comes in. Think of the constraints as boundaries that create a feasible region – a space where all the constraints are satisfied. The vertices are the corners of this region. The fundamental theorem of linear programming tells us that the maximum (or minimum) value of a linear function, like our profit function, always occurs at one of these vertices. That's why evaluating the profit function at each vertex is so crucial.
We will need to identify the vertices of the feasible region. These vertices are the points where the boundary lines intersect. In this case, we are given three vertices: (8, 1), (14, 1), and (3, 6). These points are the corners of our feasible region, and one of them will give us the maximum profit. Understanding this concept is key to mastering linear programming and evaluating profit functions effectively.
Evaluating Profit at Vertex (8, 1)
Let's start by evaluating profit function at the vertex (8, 1). This means we'll substitute x = 8 and y = 1 into our profit function:
Substitute x = 8 and y = 1:
Now, let's break this down step by step. First, we calculate the individual terms:
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- 04(8) = 0.32
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- 05(1) = 0.05
- 16 - 8 - 1 = 7
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- 06(7) = 0.42
Now, we add these values together:
So, at the vertex (8, 1), the profit P is $0.79. This means that if we produce 8 units of the first product (x) and 1 unit of the second product (y), our profit will be $0.79, considering the constraint represented by the third term in the profit function. Keep in mind, this is just the profit at one vertex. We need to do the same for the other vertices to see which one gives us the highest profit. Evaluating profit at each vertex is a critical step in finding the optimal solution. This methodical approach ensures we don't miss the maximum profit by only looking at one point.
Evaluating Profit at Vertex (14, 1)
Next up, let's evaluate profit function at the vertex (14, 1). Again, we'll substitute x = 14 and y = 1 into our profit function:
Substitute x = 14 and y = 1:
Time to break it down:
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- 04(14) = 0.56
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- 05(1) = 0.05
- 16 - 14 - 1 = 1
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- 06(1) = 0.06
Now, add 'em up:
At the vertex (14, 1), the profit P is $0.67. This tells us that producing 14 units of the first product and 1 unit of the second product results in a profit of $0.67, taking into account the constraint. It’s lower than the profit we calculated at (8, 1), but we're not done yet! We still need to evaluate profit function at the last vertex to get the full picture. This step-by-step evaluation of profit ensures we're thorough in our analysis.
Evaluating Profit at Vertex (3, 6)
Finally, we'll evaluate profit function at the vertex (3, 6). We substitute x = 3 and y = 6 into our profit function:
Substitute x = 3 and y = 6:
Let's calculate each term:
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- 04(3) = 0.12
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- 05(6) = 0.30
- 16 - 3 - 6 = 7
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- 06(7) = 0.42
Now, sum it up:
At the vertex (3, 6), the profit P is $0.84. So, producing 3 units of the first product and 6 units of the second product gives us a profit of $0.84. This is the highest profit we've seen so far! By evaluating the profit function at all vertices, we've now identified the maximum profit. This meticulous approach is the heart of linear programming, helping us find the best possible outcome.
Conclusion
Okay, guys, we've done it! We've evaluated the profit function at each vertex and found the profit at each point. Let’s recap our findings:
- At vertex (8, 1), the profit P = $0.79
- At vertex (14, 1), the profit P = $0.67
- At vertex (3, 6), the profit P = $0.84
So, what does this all mean? Well, it tells us that the maximum profit occurs at the vertex (3, 6), where the profit is $0.84. This means that to maximize profit, we should produce 3 units of the first product (x) and 6 units of the second product (y), while still adhering to the constraint in our profit function.
This process of evaluating profit functions at vertices is super powerful in linear programming. It allows us to find the optimal solution in various real-world scenarios, from production planning to resource allocation. Remember, the key is to systematically evaluate each vertex to ensure you find the absolute maximum (or minimum) value. And that’s how we conquer profit optimization! Keep practicing, and you'll become a pro at evaluating profit at each vertex in no time!