Rectangle: Length, Width, Area, And Perimeter Explained

Have you ever encountered a math problem that seemed like a puzzle? This article will guide you through solving a classic geometry problem involving rectangles, their dimensions, area, and perimeter. We'll break down each step in a clear, friendly way, so you can confidently tackle similar challenges in the future. Let's dive in!

Decoding the Rectangle Problem

Let's get started by carefully considering the problem. The core of the problem states: the length of a rectangle is five times its width, and the area of the rectangle is 245 square yards. The ultimate goal is to find the perimeter of this rectangle. This problem touches on fundamental concepts in geometry, specifically relating to rectangles, area, and perimeter. It's a perfect example of how mathematical relationships can be used to solve real-world problems.

Breaking Down the Given Information

To solve this, we first need to dissect the information provided. The relationship between the length and width is crucial, and it tells us that the length is five times the width. This can be expressed algebraically, which will be key to solving the problem. We also know the area of the rectangle, which is another critical piece of information. Remember, the area of a rectangle is calculated by multiplying its length and width. By understanding these givens, we can start formulating a plan to find the unknowns.

The Goal: Finding the Perimeter

It’s important to keep the end goal in mind: we need to determine the perimeter of the rectangle. The perimeter is the total distance around the rectangle, which is the sum of all its sides. Knowing this, we understand that we need to find both the length and the width before we can calculate the perimeter. This gives us a clear path forward and helps us organize our thoughts.

Setting Up the Equations

Now, let's translate the word problem into mathematical language. This involves using variables and formulas, which are the tools we use to express relationships in algebra. By setting up the equations correctly, we can solve for the unknowns and ultimately find the perimeter.

Defining Variables for Width and Length

The first step is to assign variables. Let's use 'w' to represent the width of the rectangle and 'l' to represent its length. This is a standard practice in algebra and helps us keep track of the different quantities. Since the length is five times the width, we can write this relationship as an equation: l = 5w. This equation is a cornerstone of our solution, as it directly connects the two dimensions of the rectangle.

Expressing the Area in Terms of Width

We know that the area of a rectangle is calculated by multiplying its length and width. The problem states that the area is 245 square yards. So, we can write the equation: Area = l * w = 245. But we also know that l = 5w. Substituting this into the area equation gives us: (5w) * w = 245. This simplifies to 5w² = 245. Now we have a single equation with one variable, which we can solve.

The Perimeter Formula

Finally, let's recall the formula for the perimeter of a rectangle. The perimeter (P) is the sum of all the sides, which can be expressed as: P = 2l + 2w. Once we find the values of 'l' and 'w', we can plug them into this formula to calculate the perimeter. This equation is our final destination, the key to answering the problem.

Solving for Width and Length

With our equations set up, we're ready to roll up our sleeves and solve for the unknown dimensions of the rectangle. This is where our algebraic skills come into play. By carefully manipulating the equations, we can isolate the variables and find their values.

Calculating the Width

From our previous setup, we have the equation 5w² = 245. To solve for 'w', we first need to isolate w². Divide both sides of the equation by 5: w² = 245 / 5 = 49. Now we have w² = 49. To find 'w', we take the square root of both sides: w = √49. The square root of 49 is 7, so the width (w) = 7 yards. Remember, when dealing with real-world measurements, we only consider the positive root.

Determining the Length

Now that we know the width, we can easily find the length. We have the equation l = 5w. Substituting w = 7 yards into this equation, we get: l = 5 * 7 = 35 yards. So, the length (l) of the rectangle is 35 yards. We’ve now successfully found both dimensions of the rectangle!

Calculating the Perimeter

With the width and length in hand, we're in the home stretch. All that remains is to use the perimeter formula to find the total distance around the rectangle. This final calculation will give us the answer we've been working towards.

Plugging in the Values

Recall the perimeter formula: P = 2l + 2w. We know that l = 35 yards and w = 7 yards. Substituting these values into the formula, we get: P = 2 * 35 + 2 * 7. Now it's just a matter of performing the arithmetic.

The Final Calculation

Let's break down the calculation step by step: 2 * 35 = 70, and 2 * 7 = 14. So, P = 70 + 14. Adding these together, we get: P = 84 yards. Therefore, the perimeter of the rectangle is 84 yards. We've successfully solved the problem!

Reviewing the Solution

Before we wrap up, let's take a moment to look back at the steps we took and ensure our solution makes sense. This is an important habit in problem-solving, as it helps us catch any potential errors and solidify our understanding.

Checking for Consistency

We found that the width is 7 yards and the length is 35 yards. Does this fit the initial condition that the length is five times the width? Yes, 35 is indeed five times 7. We also know the area is 245 square yards. Is the product of our calculated length and width equal to 245? Let's check: 35 * 7 = 245. This confirms that our dimensions are consistent with the given information. Finally, we calculated the perimeter to be 84 yards. This seems reasonable given the dimensions we found.

Key Takeaways

This problem demonstrates the power of translating word problems into mathematical equations. By defining variables, setting up equations, and carefully solving them, we can find the answers we seek. Remember to always check your solution against the original problem to ensure it makes sense. This step-by-step approach can be applied to a wide range of geometry and algebra problems.

Conclusion

Solving geometry problems involving rectangles, area, and perimeter doesn't have to be daunting. By breaking down the problem into smaller, manageable steps, setting up equations, and carefully solving them, you can confidently find the solutions. Remember to always check your work and ensure your answers make sense in the context of the problem. Now you're equipped to tackle similar challenges with ease!

For further learning and practice on geometry concepts, visit Khan Academy's Geometry Section.