Finding The Ordered Pair That Minimizes C = 60x + 85y
Hey guys! Let's dive into a fun math problem where we need to figure out which ordered pair makes the objective function C = 60x + 85y as small as possible. We've got a few options: (0,160), (55,70), (80,50), and (170,0). Sounds like a cool challenge, right? Let's break it down and find the answer together.
Understanding Objective Functions
Before we jump into the calculations, let's make sure we're all on the same page about what an objective function is. In simple terms, an objective function is a mathematical expression that we want to either maximize (make as big as possible) or minimize (make as small as possible). In our case, the objective function is C = 60x + 85y, and we're aiming to minimize it. This means we want to find the values of x and y that give us the smallest possible value for C.
The variables x and y usually represent some real-world quantities, like the number of items produced or the amount of resources used. The coefficients (60 and 85 in our case) represent the cost or profit associated with each unit of x and y. So, minimizing the objective function could mean minimizing the cost, while maximizing it could mean maximizing the profit.
Ordered pairs, like the ones we have (0,160), (55,70), (80,50), and (170,0), give us specific values for x and y. To find the ordered pair that minimizes our objective function, we need to plug these values into the equation and see which one gives us the smallest C. This is a straightforward process, but it's crucial to understand the underlying concept. We're essentially testing different scenarios to see which one results in the lowest cost, based on our objective function.
Remember, the goal is to find the combination of x and y that makes C as small as it can be. This type of problem often appears in fields like linear programming, where we're trying to optimize a certain outcome (like cost or profit) subject to some constraints. So, mastering this skill can be super useful in various real-world applications. Now, let's get to the calculations and find that minimizing ordered pair!
Plugging in the Ordered Pairs
Alright, let's get our hands dirty and start plugging in those ordered pairs into our objective function, C = 60x + 85y. This is where the rubber meets the road, and we see which pair gives us the smallest value for C. We'll go through each option one by one, so you can follow along easily. Think of it as a mini-experiment where we're testing different inputs to see the output.
First up, we have the ordered pair (0,160). This means x = 0 and y = 160. Let's substitute these values into the objective function:
C = 60(0) + 85(160) = 0 + 13600 = 13600
So, when x is 0 and y is 160, C comes out to be 13600. That's our benchmark for the first pair. Now, let's move on to the next one, (55,70). Here, x = 55 and y = 70. Plugging these values in, we get:
C = 60(55) + 85(70) = 3300 + 5950 = 9250
Okay, with (55,70), C is 9250, which is smaller than our previous result. This is interesting! It looks like we're heading in the right direction. Let's keep going and see what the next ordered pair gives us. Next, we have (80,50), so x = 80 and y = 50. Let's calculate C:
C = 60(80) + 85(50) = 4800 + 4250 = 9050
Wow, 9050! This is even smaller than 9250. It seems like (80,50) might be a strong contender for the ordered pair that minimizes C. But we can't jump to conclusions yet. We have one more pair to test: (170,0). In this case, x = 170 and y = 0. Let's plug these in:
C = 60(170) + 85(0) = 10200 + 0 = 10200
Alright, 10200 for (170,0). Now we have all the values of C for each ordered pair. Let's compare them and see which one is the smallest. Remember, we're looking for the ordered pair that minimizes C, so the smallest value wins! This step-by-step approach helps us avoid mistakes and ensures we're making the right decision based on the calculations. So, let's move on to the comparison and find our winner!
Comparing the Results
Alright, now for the exciting part! We've done the heavy lifting of plugging in all the ordered pairs into our objective function, C = 60x + 85y. Now it's time to line up the results and see which one gives us the absolute minimum value for C. This is where we put on our detective hats and analyze the data to find the best solution.
Let's quickly recap the values we got for C with each ordered pair:
- For (0,160), C = 13600
- For (55,70), C = 9250
- For (80,50), C = 9050
- For (170,0), C = 10200
Now, let's put these values side-by-side and compare them. We're looking for the smallest number, so it should be pretty clear which ordered pair minimizes C. Just a quick glance tells us that 9050 is the smallest value among all the results. This means that the ordered pair (80,50) gives us the minimum value for our objective function.
So, there you have it! By plugging in each ordered pair and comparing the resulting values of C, we've successfully identified the ordered pair that minimizes the objective function. This process might seem simple, but it's a fundamental technique in optimization problems. It's all about systematically testing different options and comparing the results to find the best one. This skill is super valuable in various fields, from business and economics to engineering and computer science. Now that we've found our minimizing ordered pair, let's summarize our findings and make sure we've got a solid understanding of the whole process.
Conclusion: The Minimizing Pair
Okay, guys, let's wrap things up and celebrate our victory! We set out to find the ordered pair that minimizes the objective function C = 60x + 85y, and we've successfully cracked the code. Through a methodical process of plugging in each ordered pair – (0,160), (55,70), (80,50), and (170,0) – and comparing the resulting values of C, we've pinpointed the winner.
As we saw, the ordered pair (80,50) gave us the smallest value for C, which was 9050. This means that when x = 80 and y = 50, our objective function is at its minimum. This is a fantastic result, and it showcases the power of using a systematic approach to solve optimization problems. We didn't just guess; we calculated, compared, and concluded with confidence.
This type of problem, where we're trying to minimize or maximize an objective function, is super common in various real-world scenarios. Imagine a business trying to minimize its costs or maximize its profits, or an engineer trying to design a structure that uses the least amount of material. These are all situations where the principles we've used today can be applied.
So, the key takeaway here is that by carefully evaluating different options and comparing their outcomes, we can make informed decisions and find the optimal solution. The process of plugging in values, calculating results, and comparing them is a powerful tool in the world of mathematics and beyond. And remember, the ordered pair (80,50) is our champion – the one that minimizes C = 60x + 85y! Great job, everyone, for working through this problem together. Keep practicing, and you'll become optimization pros in no time!
