Rectangle Area: Find The Equation For The Sides
Hey guys! Ever tackled a geometry problem where you need to figure out the dimensions of a rectangle based on its area and a relationship between its length and width? It can be a bit of a brain-bender, but let's break it down step by step. Today, we’re diving into a classic problem: A rectangle has an area of 184 square meters. The length of this rectangle is three times its width, but then we subtract 1. The big question is, what equation can we use to find the actual length and width of this rectangle?
Setting Up the Problem
So, the key here is to translate the word problem into a mathematical equation. Let's use 'x' to represent the width of the rectangle. According to the problem, the length is "three times its width diminished by 1." In algebra speak, that's 3x - 1. Now, we know that the area of a rectangle is simply length times width. And we also know that the area is 184 square meters. Putting all this together, we get our equation: x * (3x - 1) = 184. This equation is a quadratic equation, and solving it will give us the value of 'x', which is the width. Once we have the width, we can easily find the length by plugging 'x' back into the expression 3x - 1.
The Correct Equation
Based on our setup, the correct equation to find the value of each side of the rectangle is indeed x * (3x - 1) = 184. This equation accurately represents the relationship between the length, width, and area of the rectangle as described in the problem. When you expand this equation, you get 3x² - x = 184, which can be rearranged to the standard quadratic form 3x² - x - 184 = 0. Solving this quadratic equation will provide the value of 'x', which is the width of the rectangle. From there, you can calculate the length by substituting the value of 'x' into the expression 3x - 1. This approach ensures you find the correct dimensions that satisfy the given conditions of the problem.
Why This Matters
Understanding how to translate word problems into algebraic equations is a fundamental skill in mathematics. It's not just about solving for 'x'; it's about learning how to represent real-world scenarios using mathematical models. This skill is incredibly useful in various fields, including engineering, physics, economics, and computer science. Moreover, mastering this skill enhances your problem-solving abilities and logical thinking, which are valuable assets in any career or situation. So, by practicing and understanding these types of problems, you're not just acing your math test – you're building a foundation for future success. Keep practicing, and you'll become a pro at setting up and solving these equations!
Solving the Quadratic Equation
Okay, now that we've got our equation, 3x² - x - 184 = 0, let's talk about how to solve it. There are a few ways to tackle quadratic equations, but the most common methods are factoring, completing the square, and using the quadratic formula. Factoring involves finding two binomials that multiply together to give you the quadratic equation. Completing the square involves manipulating the equation to create a perfect square trinomial. And the quadratic formula is a general formula that works for any quadratic equation. In this case, the quadratic formula is probably the easiest and most reliable method. The quadratic formula is:
x = (-b ± √(b² - 4ac)) / (2a)
In our equation, a = 3, b = -1, and c = -184. Plugging these values into the quadratic formula, we get:
x = (1 ± √((-1)² - 4 * 3 * -184)) / (2 * 3)
x = (1 ± √(1 + 2208)) / 6
x = (1 ± √2209) / 6
x = (1 ± 47) / 6
This gives us two possible solutions for x:
x = (1 + 47) / 6 = 48 / 6 = 8
x = (1 - 47) / 6 = -46 / 6 = -7.67
Since the width of a rectangle cannot be negative, we discard the negative solution. Therefore, the width of the rectangle is 8 meters.
Finding the Length
Now that we know the width of the rectangle is 8 meters, we can find the length using the expression we derived earlier: length = 3x - 1. Plugging in x = 8, we get:
length = 3 * 8 - 1 = 24 - 1 = 23 meters
So, the length of the rectangle is 23 meters.
Checking Our Work
It's always a good idea to check your work to make sure your answer is correct. We can do this by multiplying the length and width to see if we get the area of the rectangle:
area = length * width = 23 * 8 = 184 square meters
Since the area we calculated matches the area given in the problem, we can be confident that our answer is correct. The width of the rectangle is 8 meters, and the length is 23 meters.
Real-World Applications
You might be wondering, "When am I ever going to use this in real life?" Well, understanding how to calculate the dimensions of rectangles (and other shapes) is essential in many fields. Architects and engineers use these calculations to design buildings and structures. Interior designers use them to plan layouts and arrange furniture. Landscapers use them to design gardens and outdoor spaces. Even everyday tasks like measuring a room for new flooring or determining how much fencing you need for a yard require these skills. So, while it might seem like a purely theoretical exercise, knowing how to solve these types of problems can be incredibly practical.
Tips for Solving Similar Problems
Here are a few tips to keep in mind when tackling similar problems:
- Read the problem carefully: Make sure you understand what the problem is asking before you start trying to solve it.
- Draw a diagram: Visualizing the problem can often make it easier to understand.
- Define your variables: Clearly define what each variable represents.
- Write an equation: Translate the word problem into a mathematical equation.
- Solve the equation: Use appropriate methods to solve the equation for the unknown variable.
- Check your work: Make sure your answer makes sense in the context of the problem.
By following these tips, you'll be well-equipped to solve a wide range of geometry and algebra problems. Keep practicing, and you'll become a math whiz in no time!
Conclusion
Alright, to wrap things up, we successfully found the equation needed to determine the sides of a rectangle with a given area and a specific relationship between its length and width. Remember, the key is to translate the word problem into a mathematical equation, solve for the unknown variable, and then use that information to find the other dimensions. And don't forget to check your work to make sure your answer is correct. With practice, you'll become a pro at solving these types of problems. Keep up the great work, and happy problem-solving!