Calculating Park Perimeter And Tree Planting Costs

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Hey guys! Let's dive into a fun math problem that involves a hexagonal park, some tree planting, and a bit of money. We're going to break down how to figure out the perimeter of the park, how many trees we need, and finally, how much all those trees will cost. It's all pretty straightforward, and I'll walk you through it step by step. So, grab your calculators, and let's get started!

Understanding the Problem: The Hexagonal Park

First off, we have a park shaped like a hexagon. Now, what's a hexagon? Well, it's a shape with six sides. In this case, the park's sides aren't all the same length, which makes things a little more interesting. We've got some sides that are 300 meters long and others that are 420 meters long. Here's a quick look at the given information:

  • Two sides are 420 meters each.
  • Four sides are 300 meters each.

Our goal is to find out the total distance around the park (the perimeter) and then figure out how many trees we can plant around it, given that we want to plant them at equal intervals and at each corner. Once we know how many trees we need, we can calculate the total cost, knowing that each tree costs 100 TL. Sounds good, right? Let's get to it!

Calculating the Perimeter

The perimeter is the total length of all the sides of the park added together. Since we have different side lengths, we need to add them up carefully. We have two sides that are 420 meters and four sides that are 300 meters. So, the calculation is as follows:

  • Perimeter = (2 × 420 m) + (4 × 300 m)
  • Perimeter = 840 m + 1200 m
  • Perimeter = 2040 m

So, the perimeter of the park is a whopping 2040 meters! That's a lot of ground to cover, or in our case, a lot of space to plant trees around.

Determining the Tree Planting Intervals and Number of Trees

Alright, now we need to figure out how far apart to plant the trees. The problem tells us that the trees need to be planted at equal intervals and at each corner of the park. This means that the distance between each tree must divide all the side lengths evenly. To find the optimal spacing, we need to find the greatest common divisor (GCD) of the side lengths. The side lengths are 300 m and 420 m. Let's break it down:

Finding the Greatest Common Divisor (GCD)

The GCD is the largest number that divides both 300 and 420 without leaving a remainder. Here’s how we can find it:

  1. Prime Factorization: Break down each number into its prime factors.
    • 300 = 2 × 2 × 3 × 5 × 5
    • 420 = 2 × 2 × 3 × 5 × 7
  2. Identify Common Factors: Look for the factors that both numbers share.
    • Both have 2, 2, 3, and 5.
  3. Multiply the Common Factors: Multiply these common factors together.
    • GCD = 2 × 2 × 3 × 5 = 60

So, the GCD of 300 and 420 is 60. This means we can plant trees every 60 meters along the perimeter. This ensures that the trees are equally spaced and that we have a tree at each corner.

Calculating the Number of Trees

Now that we know the spacing, we can figure out how many trees we need. We divide the length of each side by 60 m, and then we need to account for the corners:

  • For the 420 m sides: 420 m / 60 m = 7 intervals
  • For the 300 m sides: 300 m / 60 m = 5 intervals

Since we have two 420 m sides and four 300 m sides, and we want trees at each corner, let's calculate the total number of trees:

  • Trees on the 420 m sides: 2 sides × 7 intervals/side = 14 trees
  • Trees on the 300 m sides: 4 sides × 5 intervals/side = 20 trees

However, each corner counts as a tree, and the trees at the corners are already included in our calculations. Let's double-check. The perimeter is 2040 m. If we plant trees every 60 m, we'd have 2040 m / 60 m = 34 intervals. Since it’s a closed shape, the number of trees equals the number of intervals, so we need 34 trees in total.

Calculating the Total Cost of the Trees

Now for the final step: calculating the total cost. We know we need 34 trees, and each tree costs 100 TL. So, the calculation is simple:

  • Total Cost = Number of Trees × Cost per Tree
  • Total Cost = 34 trees × 100 TL/tree
  • Total Cost = 3400 TL

So, the total cost to plant trees around the park is 3400 TL. Pretty neat, huh?

Conclusion: Putting It All Together

Alright, guys, we've done it! We've successfully calculated the perimeter of the hexagonal park, figured out the optimal spacing for the trees, determined the total number of trees needed, and calculated the total cost. Here's a quick recap:

  • Perimeter: 2040 meters
  • Tree Spacing: 60 meters
  • Number of Trees: 34
  • Total Cost: 3400 TL

This was a fun problem that combined geometry with a little bit of practical application. Understanding how to find the perimeter, use the GCD, and calculate costs is useful in all sorts of real-world scenarios. Keep practicing, and you'll become math wizards in no time! Thanks for joining me today; I hope you enjoyed the ride!