Garden Area Calculation: A Geometry Problem

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Hey guys! Let's dive into a fun geometry problem where we'll calculate the area of a rectangular garden. This is a classic question that combines perimeter, side lengths, and the concept of coprime numbers. So, let's break it down step by step to make sure we understand everything clearly. Understanding the problem thoroughly is the first and most crucial step in solving any mathematical question. In this article, we’ll explore how to tackle this problem, making it super easy to grasp and apply similar concepts in the future.

Understanding the Basics of Perimeter and Area

First, let's brush up on some basics. The perimeter of a rectangle is the total distance around its sides. If we have a rectangle with length l and width w, the perimeter P is given by the formula:

P = 2(l + w)

The area, on the other hand, is the space enclosed within the rectangle. The area A is calculated by multiplying the length and the width:

A = l * w*

Now, let's talk about coprime numbers. Coprime numbers, also known as relatively prime numbers, are numbers that have no common factors other than 1. For example, 8 and 15 are coprime because their only common factor is 1. This concept will be important in finding the possible side lengths of our garden.

Applying the Formulas to Our Problem

In our problem, we know the perimeter P is 24 meters. So, we can set up the equation:

24 = 2(l + w)

Divide both sides by 2:

12 = l + w

This tells us that the sum of the length and width of the garden is 12 meters. Our task now is to find pairs of numbers that add up to 12 and are coprime (and also greater than 1, as stated in the problem).

Finding the Coprime Side Lengths

Let's list the pairs of numbers that add up to 12:

  • 1 + 11
  • 2 + 10
  • 3 + 9
  • 4 + 8
  • 5 + 7
  • 6 + 6

However, we need to consider that the side lengths must be greater than 1 meter. So, we can eliminate the pair 1 + 11. Now, let's check which of the remaining pairs are coprime:

  • 2 and 10: These have a common factor of 2, so they are not coprime.
  • 3 and 9: These have a common factor of 3, so they are not coprime.
  • 4 and 8: These have a common factor of 4, so they are not coprime.
  • 5 and 7: These have no common factors other than 1, so they are coprime.
  • 6 and 6: These have a common factor of 6, so they are not coprime.

Therefore, the only pair of coprime numbers that fit our condition is 5 and 7. So, the length and width of the garden are 5 meters and 7 meters.

Calculating the Area

Now that we know the length (l) and width (w) of the garden, we can calculate the area A using the formula:

A = l * w*

Plug in the values:

A = 5 * 7

A = 35 square meters

So, the area of the garden is 35 square meters.

Choosing the Correct Answer

Looking at the options provided:

  • A) 20
  • B) 27
  • C) 36
  • D) 35

The correct answer is D) 35. We've successfully found the area of the garden by understanding the concepts of perimeter, area, and coprime numbers. Always double-check your calculations to make sure you haven't made any simple mistakes.

Why This Problem Matters

This type of problem is more than just a math exercise. It helps us develop critical thinking and problem-solving skills. By breaking down a complex problem into smaller, manageable steps, we can find the solution more easily. Plus, understanding these geometrical concepts is super practical in real life, whether you're planning a garden or designing a room.

Real-World Applications of Geometry

Geometry isn't just confined to textbooks; it's all around us! Here are a few ways geometrical concepts are used in the real world:

  1. Architecture: Architects use geometry to design buildings, ensuring they are structurally sound and aesthetically pleasing. Concepts like symmetry, angles, and spatial relationships are crucial.
  2. Engineering: Engineers use geometry to design bridges, roads, and other infrastructure. Understanding shapes and their properties helps in creating safe and efficient designs.
  3. Urban Planning: City planners use geometry to design layouts for cities, including streets, parks, and buildings. Efficient use of space is key in urban development.
  4. Interior Design: Interior designers use geometry to arrange furniture and decor in a room, creating harmonious and functional spaces. Symmetry, balance, and spatial awareness are essential.
  5. Computer Graphics: Geometry is the backbone of computer graphics and animation. Creating 3D models and virtual environments relies heavily on geometrical principles.
  6. Navigation: GPS systems and maps use geometry to determine locations and plan routes. Understanding coordinates and distances is fundamental in navigation.
  7. Fashion Design: Fashion designers use geometry to create patterns and construct clothing. The fit and drape of a garment depend on geometrical shapes and measurements.

Tips for Solving Similar Problems

To ace similar problems, here are some tips to keep in mind:

  1. Understand the Definitions: Make sure you have a solid grasp of the definitions of perimeter, area, and coprime numbers. Knowing the basics is half the battle.
  2. Break Down the Problem: Divide the problem into smaller, more manageable steps. This makes it less daunting and easier to solve.
  3. Write Down the Formulas: Write down the relevant formulas before you start solving. This helps you stay organized and reduces the chances of making mistakes.
  4. List the Possibilities: When dealing with coprime numbers or other similar constraints, list all the possibilities and then narrow them down based on the given conditions.
  5. Check Your Answer: Always check your answer to make sure it makes sense in the context of the problem. Did you calculate the area in the correct units? Does the answer seem reasonable?

Conclusion: Mastering Geometry is Fun!

So, there you have it! We've successfully solved a geometry problem involving the perimeter and area of a rectangular garden, along with the concept of coprime numbers. Remember, practice makes perfect, so keep working on similar problems to sharpen your skills. Geometry might seem challenging at first, but with a bit of effort and the right approach, it can actually be quite fun!

Understanding these concepts can be incredibly useful in everyday life, from planning your garden to designing your living space. Keep exploring and learning, guys! You've got this!