Maximum Tree Spacing For A 36m X 48m Rectangular Field

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Let's dive into an interesting math problem about planting trees around a rectangular field! We'll break down the problem step by step, making sure you understand every detail. So, if you're ready, let's get started!

Understanding the Problem

Okay, guys, so here's the deal: Imagine we have a rectangular field. This field has sides of 36 meters and 48 meters. We want to plant trees around this field, but there are a few rules we need to follow:

  • Trees at the Corners: We absolutely have to plant trees at each corner of the rectangle.
  • Equal Spacing: The distance between any two trees must be the same.
  • Natural Number Distance: This distance must be a whole number (like 1, 2, 3, etc.) in meters. No fractions or decimals allowed!

The big question we need to answer is: What is the maximum possible distance between the trees? Basically, we want to space them out as much as we can while still following all the rules.

This is a classic math problem that combines geometry and number theory. To solve it, we need to figure out what spacing will work perfectly for both the 36-meter side and the 48-meter side. Think of it like fitting puzzle pieces together – the tree spacing has to fit evenly into both lengths.

Finding the Greatest Common Divisor (GCD)

To figure out the maximum distance, we need to find something called the Greatest Common Divisor (GCD), also sometimes referred to as the Highest Common Factor (HCF). Don't let the name scare you! It sounds complicated, but it's actually pretty simple.

The GCD of two numbers is the largest number that divides both of them perfectly, without leaving any remainder. In our case, we need to find the GCD of 36 and 48. This GCD will be the maximum distance we can have between the trees.

There are a couple of ways to find the GCD. Let's look at two common methods:

Method 1: Listing the Factors

  1. List the factors of 36: The factors of 36 are the numbers that divide 36 evenly. They are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
  2. List the factors of 48: The factors of 48 are the numbers that divide 48 evenly. They are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.
  3. Identify common factors: Look for the numbers that appear in both lists. The common factors of 36 and 48 are 1, 2, 3, 4, 6, and 12.
  4. Find the greatest: From the common factors, the largest one is 12. So, the GCD of 36 and 48 is 12.

Method 2: Prime Factorization

  1. Prime factorize 36: Break down 36 into its prime factors. This means writing 36 as a product of prime numbers (numbers that are only divisible by 1 and themselves). The prime factorization of 36 is 2 x 2 x 3 x 3, which can also be written as 2² x 3².
  2. Prime factorize 48: Do the same for 48. The prime factorization of 48 is 2 x 2 x 2 x 2 x 3, or 2⁴ x 3.
  3. Identify common prime factors: Look for the prime factors that both numbers share. Both 36 and 48 have 2 and 3 as prime factors.
  4. Multiply the lowest powers of common prime factors: For each common prime factor, take the lowest power that appears in either factorization. We have 2² (from 36) and 2⁴ (from 48), so we take 2². We also have 3² (from 36) and 3 (from 48), so we take 3. Multiply these together: 2² x 3 = 4 x 3 = 12. So, the GCD of 36 and 48 is 12.

No matter which method you use, you'll find that the GCD of 36 and 48 is 12. This means the maximum distance between the trees can be 12 meters.

Calculating the Number of Trees

Now that we know the maximum distance between the trees (12 meters), let's figure out how many trees we'll need to plant around the field. To do this, we'll calculate the number of trees needed for each side and then add them up.

Trees on the 36-meter Side

  • Divide the length of the side by the distance between trees: 36 meters / 12 meters/tree = 3 trees
  • This means we'll have trees at the beginning, middle, and end of the 36-meter side. So, we need 3 trees on each of the two 36-meter sides, but we need to be careful not to double-count the corner trees.

Trees on the 48-meter Side

  • Divide the length of the side by the distance between trees: 48 meters / 12 meters/tree = 4 trees
  • This means we'll have trees evenly spaced along the 48-meter side, including one at each end. So, we need 4 trees on each of the two 48-meter sides, again being mindful of not double-counting the corners.

Total Number of Trees

Here's where it gets a little tricky. If we simply added up the trees on each side (3 + 3 + 4 + 4), we'd be counting the corner trees twice. There are a couple of ways to avoid this:

Method 1: Subtract the Overlap

  • We have 3 trees on each 36-meter side, but we've counted two of them (the corners) twice. So, we effectively have 3 trees per side.
  • We have 4 trees on each 48-meter side, but again, we've counted two corner trees twice, so we effectively have 4 trees per side.
  • Add the unique trees from each side: (3 - 2) + (3 - 2) + (4 - 2) + (4 - 2) = 1 + 1 + 2 + 2 = 6 trees. However, we initially subtracted the corner trees, so we need to add them back in: 6 + 4 (corner trees) = 12 trees.

Method 2: Calculate the Perimeter First

  • The perimeter of the rectangle is 2 * (36 meters + 48 meters) = 2 * 84 meters = 168 meters.
  • Divide the total perimeter by the distance between trees: 168 meters / 12 meters/tree = 14 trees.

Both methods give us the same answer: we need a total of 14 trees to plant around the field with a maximum spacing of 12 meters.

Summarizing the Solution

Okay, let's recap what we've done:

  1. Understood the Problem: We needed to find the maximum distance between trees planted around a rectangular field with specific constraints.
  2. Found the GCD: We determined that the greatest common divisor of 36 and 48 is 12. This is the maximum distance between the trees.
  3. Calculated the Number of Trees: We figured out that we need 14 trees in total to plant around the field, ensuring there's a tree at each corner and the spacing is 12 meters.

So, there you have it! We've successfully solved the problem. The maximum distance between the trees is 12 meters, and you'll need 14 trees to complete the task.

Real-World Applications and Further Exploration

This type of problem isn't just a math exercise; it has real-world applications. For example, landscape architects or farmers might use this kind of calculation to plan the spacing of plants or trees in a field or garden. It helps them ensure that resources like sunlight and water are distributed evenly.

If you're interested in exploring further, you could try changing the dimensions of the field and recalculating the maximum tree spacing and the number of trees needed. You could also investigate other factors that might influence tree spacing in a real-world scenario, such as the size of the trees or the type of soil.

Conclusion

Math problems like this one might seem tricky at first, but by breaking them down into smaller steps and using the right tools (like the GCD), they become much more manageable. Keep practicing, and you'll become a math whiz in no time! Remember, math is all around us, and understanding it helps us solve real-world problems and make informed decisions. Keep exploring and keep learning, guys!