Fencing A 25-Acre Field: A Math Problem!

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Fencing a 25-Acre Field: A Math Problem!

Hey guys! Let's dive into a fun math problem that's super practical. Imagine you're tasked with fencing a massive 25-acre rectangular field. One side of the field is exactly 1/4 mile long. The big question is: How much fencing do you actually need? We'll break it down step-by-step so you can totally nail this! This problem is a classic example of applying math to real-world scenarios. We'll need to remember some key conversions, understand how to calculate the perimeter of a rectangle, and then crunch some numbers. Ready? Let's get started!

Understanding the Problem: The Basics

Okay, so we've got a rectangular field, and we know a few crucial details. First, the total area of the field is 25 acres. Second, one side of the rectangle measures 1/4 of a mile. Our goal is to figure out the total length of fencing needed to completely enclose the field. That means we need to find the perimeter. Remember, the perimeter is the total distance around the outside of a shape. For a rectangle, the perimeter is calculated by adding up the lengths of all four sides. Since opposite sides of a rectangle are equal, we just need to know the length of the other side. This is where a little bit of cleverness comes in. We need to work with the area of the rectangle and the length of one side to find out the other side's length. Then, we can calculate the perimeter and convert our answer into yards, which is the standard unit for fencing. Don't worry, we'll go through each step carefully, so it's all crystal clear. This problem helps us understand how different measurements relate to each other and why units of measurement are super important.

Converting Acres to Square Miles

To begin, we need to convert the area from acres to square miles, as one side is given in miles. We know that 1 square mile is equal to 640 acres. Therefore, to convert 25 acres to square miles, we divide by 640:

25 acres / 640 acres/square mile = 0.0390625 square miles

So, the area of the field is 0.0390625 square miles. Got it?

Finding the Other Side's Length

Next, we need to determine the length of the other side of the rectangle. We know the area of a rectangle is calculated using the formula: Area = Length × Width. We know the area (0.0390625 square miles) and the length of one side (1/4 mile). Let's call the other side 'x'. We can set up the equation:

  1. 0390625 square miles = (1/4 mile) × x

To solve for 'x', we divide the area by 1/4 (or multiply by 4):

x = 0.0390625 square miles / (1/4 mile)

x = 0.0390625 square miles × 4

x = 0.15625 miles

So, the other side of the rectangle is 0.15625 miles long.

Calculating the Perimeter

Now, we can calculate the perimeter. Remember, the perimeter of a rectangle is calculated as: Perimeter = 2 × (Length + Width). In our case:

Perimeter = 2 × (1/4 mile + 0.15625 mile)

Perimeter = 2 × (0.25 mile + 0.15625 mile)

Perimeter = 2 × 0.40625 miles

Perimeter = 0.8125 miles

The perimeter of the field is 0.8125 miles.

Converting Miles to Yards

We're almost there, folks! The final step is to convert the perimeter from miles to yards. We know that 1 mile is equal to 1,760 yards. So, we multiply the perimeter in miles by 1,760:

  1. 8125 miles × 1,760 yards/mile = 1,430 yards

Therefore, you will need 1,430 yards of fencing to enclose the field completely.

The Answer and Explanation

So, the correct answer is B. 1,430. This problem beautifully demonstrates how different units of measurement relate to each other. We started with acres, converted to square miles, used the known side to calculate the other side, and then calculated the perimeter in miles before finally converting it to yards. Each step was essential, and by breaking it down this way, the solution becomes much more manageable. Understanding units and conversions is a crucial skill in math and in real life!

To recap:

  1. Converted acres to square miles. This was necessary to use the given length (in miles). Doing so allows us to relate the known information and use it to solve for unknown variables, in this case, the other side. Understanding relationships is a key concept in math.
  2. Calculated the length of the other side. We used the area formula and the known length to find the unknown side's length. This step helps in understanding how formulas are used to find missing dimensions.
  3. Calculated the perimeter. Using the calculated side lengths, we were able to figure out how much fencing we needed. The perimeter gave us the total distance to be enclosed. The concept of perimeter helps us know the outer boundary of a given shape.
  4. Converted miles to yards. This gives us our final answer in the required units, making it practical for real-world application. Conversion steps are vital when working in real-world scenarios.

By following these steps, we've successfully solved the problem! Feel confident applying this approach to similar problems. This type of problem-solving helps you think critically and logically!

More Practice and Tips

Want to get even better? Here are a few tips and some extra practice questions:

  • Practice with Different Shapes: Try this with triangles or circles to see how the formulas change.
  • Focus on Conversions: The trickiest part is often the unit conversions. Make sure you know your conversions! Use online converters to double-check your work.
  • Draw a Diagram: Always draw a diagram of the problem. This will help you visualize the problem and stay organized.
  • Check Your Units: Always make sure your units are consistent throughout the problem. A mistake in units can lead to a wrong answer.

Extra practice questions:

  1. A rectangular garden is 20 feet long and 15 feet wide. How much fencing is needed to enclose it? (Answer: 70 feet)
  2. A square field has an area of 900 square meters. What is the length of one side? What is the perimeter of this field? (Answer: 30 meters, 120 meters)
  3. A circular pond has a radius of 7 meters. What is the circumference (the distance around the pond)? (Use π ≈ 22/7) (Answer: 44 meters)

Keep practicing, and you'll become a fencing expert in no time! This is just one example of the many ways math is relevant in everyday life. Good luck and have fun!