Maximize F(x, Y): A Linear Programming Problem

Hey guys! Let's tackle this math problem. We're tasked with finding the maximum value of the function f(x, y) = 4x + 28y. But hold on, there's a catch! We're not just dealing with any old function; this one has some conditions attached. These conditions are called constraints, and they shape the playing field, dictating the valid values of x and y. To crack this, we'll dive into the world of linear programming. Ready? Let's get started!

Understanding the Constraints: The Foundation of Our Problem

First, let's break down those constraints. They're the rules of the game, if you will. They tell us where x and y are allowed to hang out.

• Constraint 1: 5x + 4y ≤ 34: This inequality describes a region in the xy-plane. Any point (x, y) that satisfies this inequality falls within that region. Think of it like a boundary that keeps our solutions in check.
• Constraint 2: 3x + 5y ≤ 30: Similar to the first constraint, this one defines another region. The points (x, y) that satisfy this inequality are also valid solutions.
• Constraint 3: x ≥ 0: This one tells us that x can't be negative. It restricts our solutions to the right side of the y-axis.
• Constraint 4: y ≥ 0: Likewise, this tells us that y can't be negative, keeping our solutions above the x-axis.

These four constraints together define a specific area where our solution must reside. This area is a feasible region or the solution space. The goal is to find the point within this region that gives us the highest possible value for f(x, y). Now, let's visualize this. Imagine these constraints as lines on a graph. The feasible region is the area enclosed by these lines and the axes. The maximum value of the function will always lie on the boundary of this feasible region, specifically at one of its corners, also called vertices. This is a fundamental concept in linear programming, and it's key to solving our problem. It's kinda like finding the highest point on a mountain range; the peak is the maximum value. We'll analyze the vertices of the feasible region to find our answer. Understanding and visualizing these constraints is essential to solving this kind of problem effectively.

Graphical Representation: Visualizing the Solution Space

To make this problem easier, let's bring in some visual aids. We're going to graph these constraints to see the feasible region. This is where things get a bit more concrete. First, we'll treat the inequalities as equalities to draw the boundary lines. For example, 5x + 4y ≤ 34 becomes 5x + 4y = 34. We can then find the x and y intercepts of the line; for x-intercept, set y = 0 and solve for x. Similarly for y-intercept, set x = 0 and solve for y. Once we have the intercepts, we can draw the line. Do this for each constraint:

• Constraint 1: 5x + 4y = 34. The x-intercept is (6.8, 0) and the y-intercept is (0, 8.5).
• Constraint 2: 3x + 5y = 30. The x-intercept is (10, 0) and the y-intercept is (0, 6).

Plot these lines on a graph. The region satisfying 5x + 4y ≤ 34 is below the line, and the region satisfying 3x + 5y ≤ 30 is below that line. Then, consider the constraints x ≥ 0 and y ≥ 0. These tell us that the feasible region is in the first quadrant (where both x and y are non-negative). The area that satisfies all these constraints is the feasible region. It's a polygon, and the corners of that polygon (vertices) are where the maximum value of our function f(x, y) is most likely to be found. This graphical method provides a clear picture of the constraints and helps identify potential solutions, making the problem much easier to solve. This visual approach helps us understand the interaction between the constraints and the function we're trying to maximize. This visual aid is crucial for a comprehensive understanding. It's like having a map to find the treasure! The graph will show us the possible values of x and y that satisfy all the conditions and helps us zero in on the solution.

Finding the Vertices: Identifying Potential Solutions

Alright, let's find those vertices. These are the corner points of our feasible region, where the constraint lines intersect. We know that the maximum value of f(x, y) will occur at one of these points. So, we need to find the coordinates of each vertex.

• Vertex 1: Intersection of x = 0 and y = 0: This is simply the origin, (0, 0).

• Vertex 2: Intersection of x = 0 and 3x + 5y = 30: Substitute x = 0 into the equation 3x + 5y = 30 gives us 5y = 30, which gives us y = 6. So this vertex is (0, 6).

• Vertex 3: Intersection of 5x + 4y = 34 and 3x + 5y = 30: We can solve this system of equations using substitution or elimination. Let's use elimination. Multiply the first equation by 3 and the second equation by 5.

  • 15x + 12y = 102
  • 15x + 25y = 150

  Subtract the first from the second, which gives us 13y = 48, so y = 48/13. Substitute y = 48/13 into the first equation: 5x + 4(48/13) = 34. This gives 5x = 34 - 192/13, so x = 150/65 = 30/13. Therefore, this vertex is (30/13, 48/13).

• Vertex 4: Intersection of y = 0 and 5x + 4y = 34: Substitute y = 0 into 5x + 4y = 34, resulting in 5x = 34, so x = 34/5. This gives us the vertex (34/5, 0).

We have found all four vertices of the feasible region. They are (0, 0), (0, 6), (30/13, 48/13), and (34/5, 0). Now, we'll plug these coordinates into our function f(x, y) = 4x + 28y to find the maximum value. We have successfully isolated the critical points where the maximum value could be found by carefully analyzing the constraint equations. The accuracy of these vertices' coordinates is paramount to finding the correct solution. The process of identifying vertices is fundamental in this kind of problem.

Evaluating f(x, y) at the Vertices: Finding the Maximum Value

Now, let's evaluate f(x, y) = 4x + 28y at each of the vertices we found. This is where we put all the pieces together and find our final answer. Remember, we're looking for the largest value.

• At (0, 0): f(0, 0) = 4(0) + 28(0) = 0
• At (0, 6): f(0, 6) = 4(0) + 28(6) = 168
• At (30/13, 48/13): f(30/13, 48/13) = 4(30/13) + 28(48/13) = 120/13 + 1344/13 = 1464/13 ≈ 112.62
• At (34/5, 0): f(34/5, 0) = 4(34/5) + 28(0) = 136/5 = 27.2

Comparing these values, the maximum value of f(x, y) is 168, which occurs at the point (0, 6). We've found the maximum value by systematically checking all the corner points of the feasible region. This step is the culmination of our work, where all previous steps converge to give us the ultimate solution. The careful calculation at each vertex is critical. The highest value is the answer we've been seeking throughout the entire process. Success! We've found the answer! It's like reaching the summit of a mountain. We have carefully computed the function's value at each vertex, ensuring our result's accuracy. This method is the heart of solving linear programming problems.

Conclusion: Our Final Answer

Therefore, the maximum value of the function f(x, y) = 4x + 28y that satisfies the constraints is 168. The correct answer is C.

We have gone from understanding the constraints, to visualizing the solution space, finding the critical points, and finally, calculating the maximum value. This systematic approach is the essence of solving a linear programming problem. Congrats, guys! We cracked the code! The key is to understand that linear programming problems, like the one we just solved, can be solved effectively by finding the vertices of a feasible region and evaluating the objective function at those points. Keep practicing, and you'll become masters of these problems in no time. This is the end of our problem-solving journey. You're now equipped with the knowledge and tools to tackle similar problems with confidence.