Maximize Square Plots: Area & Cost Calculation

Hey guys! Let's dive into a fun geometry problem. We've got a rectangular field, and our mission is to divide it into the biggest possible square plots. Then, we'll figure out the area of those squares and even calculate the cost if we sell them. Sounds good? Let's get started!

Understanding the Problem: The Rectangular Field

Okay, so we're dealing with a rectangular field that's a whopping 360 meters long and 150 meters wide. Imagine a giant rectangle out in the middle of nowhere. Our goal is to carve this rectangle up into smaller squares, all with the same size. But here's the kicker: we want these squares to be as large as possible. This means we're looking for the greatest common divisor (GCD) of the length and width to determine the side length of our squares. Why the GCD? Because it ensures that the squares fit perfectly within the rectangle without any leftover space or weird slivers.

Think of it like this: If we don't use the largest possible square size, we'll end up with smaller squares and some wasted space. We don't want that! We want to use every bit of our field efficiently. This is all about maximizing the area of the individual square plots. This problem isn't just about math; it's about optimization. It's about finding the best way to utilize the space we have. This process of dividing the field into squares is also an example of how geometry and real-world problems often intersect. The principles we're applying here are applicable in fields like land management, architecture, and even urban planning. For example, imagine a builder trying to maximize the size of apartments within a building or a farmer trying to plan a field layout. Understanding this problem is understanding the potential of efficient space division.

So, before we go any further, let's nail down these key numbers. We have a length of 360 meters and a width of 150 meters. Keep these numbers in mind – they're the building blocks of our solution. Now, let's talk about the squares themselves. Remember, our objective is to determine the dimensions of the largest possible square plots. This sets the stage for our mathematical exploration.

Finding the Greatest Common Divisor (GCD)

Alright, time to get a little mathematical, but don't worry, it's not too complicated. To find the size of the biggest square we can make, we need to find the greatest common divisor (GCD) of the length (360 meters) and the width (150 meters). The GCD is the largest number that divides both 360 and 150 without leaving any remainders. There are a couple of ways to find the GCD. We could list all the factors of both numbers and pick the biggest one they have in common. However, especially when dealing with larger numbers, that can be a bit tedious. A more efficient method is the Euclidean algorithm. This is a step-by-step process that makes things much easier.

Here's how the Euclidean algorithm works in this case: we start by dividing the larger number (360) by the smaller number (150). 360 divided by 150 is 2 with a remainder of 60. Then, we divide the previous divisor (150) by the remainder (60). 150 divided by 60 is 2 with a remainder of 30. We repeat this process. Now, we divide 60 by 30, which gives us 2 with no remainder. When the remainder is 0, the last non-zero remainder is the GCD. Therefore, the GCD of 360 and 150 is 30. This means the largest possible square plots will have sides of 30 meters each.

What does this mean for our field? It means we can divide the 360-meter length into 12 squares (360 / 30 = 12), and the 150-meter width into 5 squares (150 / 30 = 5). So, we'll have a grid of 12 squares across and 5 squares down, for a total of 60 squares (12 * 5 = 60). This is the most efficient way to divide the field, ensuring that all our plots are equal in size and maximizing the use of our entire field. The use of GCD, as you can see, is not just a math trick; it's a practical tool for efficient planning and resource management, especially in situations where uniform division is required.

Now we're one step closer to solving our problem! The determination of the GCD is the cornerstone of dividing the field into the largest possible square plots, which directly affects the area of each plot, and consequently, the final cost calculation. Let's move on to the next part and calculate the area of those perfect squares we've created.

Calculating the Area of Each Square Plot

Okay, now that we know the side length of each square plot is 30 meters, we can easily calculate its area. The area of a square is found by multiplying the side length by itself (side length * side length, or side^2). So, in our case, the area of each square plot is 30 meters * 30 meters = 900 square meters. This means each of our perfectly formed square plots covers 900 square meters of the field. This calculation is straightforward, but it's important for determining the total value of our field when we get to the selling part.

Knowing the area of each square plot is crucial because it allows us to determine the total area of the field efficiently. For example, if we didn't calculate the area of each square individually, we would have to calculate the total area of the rectangle and then determine the individual area by the number of plots. We would still arrive at the same conclusion, but the methodology would be unnecessarily cumbersome. This small calculation underscores how basic math concepts like area are directly applicable to practical scenarios.

The area calculation also offers the opportunity to explore related concepts. For example, we could discuss the implications of having larger or smaller square plots. What if we wanted to change the dimensions of the squares? How would that affect the total number of plots? How would that affect the sale price? Questions like these expand the scope of the problem and encourage deeper thinking. Also, we could introduce concepts like perimeter and how the perimeter of each square would relate to its area. See, one simple calculation can open doors to many different avenues of mathematical and practical exploration.

Now, armed with the knowledge of each square plot's area, we can progress to the final piece of the puzzle. Now, let's talk about the money! How much will we make if we sell those plots at a certain price?

Calculating the Total Selling Price

Alright, let's get down to the exciting part: the money! We know the area of each square plot (900 square meters), and we know the selling price per square meter (125). To find the selling price of each plot, we simply multiply the area of one plot by the price per square meter. So, each plot will sell for 900 square meters * $125/square meter = $112,500. Not bad, right?

To find the total selling price for the entire field, we need to multiply the selling price per plot by the total number of plots. We've already determined that we can divide the field into 60 plots. So, the total selling price is $112,500/plot * 60 plots = $6,750,000. That's a pretty sweet deal!

This final calculation is the culmination of all our work. We started with a rectangular field and through the strategic application of mathematical concepts, we arrived at a final valuation. It exemplifies the power of applying mathematical principles to practical real-world problems. The methodology we've employed can be adapted to various scenarios. For instance, consider different shapes of fields or various prices per square meter. The ability to calculate the selling price and how to maximize profit will give you a good advantage. The entire process, from finding the GCD to calculating the total selling price, demonstrates the elegance and usefulness of mathematical thinking.

Summary

So, here's a quick recap of what we've done, guys! We started with a 360-meter by 150-meter rectangular field. We wanted to divide it into the largest possible square plots, found the GCD to determine the side length of the squares (30 meters), calculated the area of each plot (900 square meters), and then figured out the total selling price of the entire field ($6,750,000). Not too shabby, huh?

This is a good example of how math isn't just about numbers; it's a tool for solving real-world problems. Whether you're planning a garden, managing a property, or even just trying to figure out the best way to divide a pizza, these mathematical principles can come in handy. Keep practicing, and you'll be a math whiz in no time!