Solving Rectangle Area Problems: Finding Length And Equations
Hey everyone! Let's dive into a classic math problem involving rectangles. We'll figure out the length of a rectangle when we know its area and how the length relates to the width. Plus, we'll explore which equation best describes the situation. Get ready to flex those problem-solving muscles! This is a super common type of question, so understanding it is a win-win. We are going to learn about rectangle area problems and how to find the length and represent the situation with an equation. Let's get started, shall we?
Understanding the Problem: Breaking it Down
Alright, let's break down the problem. We're given that the length of a rectangle is 4 more than its width. This is super important because it tells us how the two sides are connected. We can start thinking about this with an example, let's say the width is 1 yard, then the length would be 1 + 4 = 5 yards. But we don't know the actual width yet; we only know the relationship between length and width. We're also told that the area of the rectangle is 60 square yards. Remember, the area of a rectangle is found by multiplying its length and width. Finally, the big question: what is the length of the rectangle? To solve this, we'll need to use both pieces of information: the relationship between length and width, and the total area. The area is the total space inside the rectangle, kind of like how much carpet you'd need to cover the floor. Let's get into it. We'll use algebra to set up an equation and find the length.
To solve this problem effectively, we must convert the provided data into equations and numerical forms to compute the length. Understanding the core concepts, like the properties of rectangles, is essential. The length of a rectangle is 4 more than the width, and the area of the rectangle is 60 square yards; therefore, the objective is to find the length of the rectangle. Before we dive into the solution, let's review some basics about rectangles. A rectangle is a four-sided shape with opposite sides that are equal in length and all interior angles that measure 90 degrees. The area of a rectangle is calculated by multiplying its length by its width, represented by the formula: Area = Length × Width or A = l × w.
In this problem, the length of the rectangle is 4 more than the width. We can represent this relationship algebraically. If we represent the width as w, then the length l can be represented as w + 4. Because the area is given as 60 square yards, we can now substitute the l for w + 4 in the area formula. This is the key to solving the problem, as it allows us to create an equation with a single variable. When we have an equation with a single variable, we can use algebraic operations to solve for the variable and finally find the length. Thus, we're turning word problems into equations, which is the core of algebra. The goal is to isolate the variable we're trying to find, which in this case is the length.
Setting Up the Equation: Turning Words into Math
Okay, guys, let's translate the problem into an equation. We know the area (A) of a rectangle is calculated by: A = length × width. We also know the length is 4 more than the width. So, let's use 'w' to represent the width. Therefore, the length (l) is 'w + 4'. Since the area is 60 square yards, we can substitute these values into the area formula:
- 60 = (w + 4) × w
See? We've turned the word problem into a mathematical equation! Now, let's simplify and solve for 'w'. First, distribute the 'w' across the parentheses:
- 60 = w² + 4w
This is a quadratic equation. To solve it, let's rearrange it to equal zero:
- w² + 4w - 60 = 0
Now, we can solve for w using factoring, the quadratic formula, or by completing the square. Factoring is often the easiest method if the equation can be factored, so let's try that. We need to find two numbers that multiply to -60 and add up to 4. Those numbers are 10 and -6.
Therefore:
- (w + 10)(w - 6) = 0
This means either (w + 10) = 0 or (w - 6) = 0. Solving for w, we get w = -10 or w = 6. Since the width of a rectangle cannot be negative, we discard -10. So, the width (w) = 6 yards. Now, let's find the length! The length is w + 4, so the length = 6 + 4 = 10 yards.
We used the area formula in combination with the relationship between length and width. The equation setup, solving for 'w' and finding the length are key steps. Always check if your answers make sense in the real world. Does a rectangle with a width of 6 yards and a length of 10 yards sound reasonable? Yes! Its area is indeed 60 square yards (6 × 10 = 60).
Solving the Equation and Finding the Length
Let's get our hands dirty and solve that equation! As we discussed, we've got the quadratic equation w² + 4w - 60 = 0. Factoring this equation, we look for two numbers that multiply to -60 and add up to 4. We found these numbers to be 10 and -6. This allows us to rewrite the equation as (w + 10)(w - 6) = 0. The critical part here is understanding that if the product of two factors is zero, then at least one of the factors must be zero. This is the zero-product property.
So, either (w + 10) = 0 or (w - 6) = 0. Solving each of these equations separately gives us two possible solutions for the width, w = -10 or w = 6. However, since a rectangle's width cannot be a negative value, we discard the -10 solution. This leaves us with the width, w = 6 yards. Now that we've found the width, we can easily calculate the length. Remember, the length is 4 more than the width. This means the length l = w + 4 = 6 + 4 = 10 yards. Bingo! We've found both the width and the length of the rectangle. The next step is verifying the solution. To confirm the answer, multiply the length by the width and verify that we get the correct area. This will help us ensure that our calculations are correct. Therefore, the length of the rectangle is 10 yards. So, we have a width of 6 yards and a length of 10 yards, and the area is 60 square yards. Now, we have found our solution.
Checking Your Answer: Does It Make Sense?
Alright, guys, always a good idea to check your work! We found that the length of the rectangle is 10 yards and the width is 6 yards. Let's make sure this makes sense with the information we were given. First, does the length fit the description that it is 4 yards more than the width? Yes, 10 yards is indeed 4 yards more than 6 yards. Second, does the area of the rectangle come out to 60 square yards? To check this, we multiply the length by the width: 10 yards × 6 yards = 60 square yards. Yes! Our answer checks out. It fits the criteria given in the problem. Therefore, our calculations are correct.
Verifying your answer is important because it helps ensure the reasonableness of the solution. If your calculations led to a negative dimension or an area that didn't match the problem's information, you'd know something went wrong. Always take a moment to reflect on the answer and confirm that it makes sense within the context of the problem. Checking can include performing a mental calculation or using the obtained measurements in the formula to ensure the resulting value matches the given value. By going through this process, you ensure the reliability and accuracy of your solutions.
Identifying the Correct Equation
We already established the equation we used to solve the problem, so let's look at the choices. The original equation we established was 60 = (w + 4) × w which, after simplification, becomes w² + 4w - 60 = 0.
If you were given multiple-choice options, you'd be looking for an equation equivalent to this. Remember, this is the equation that represents the problem in a mathematical way. This is the core of the problem. The other options might include equations that don't correctly represent the area formula or the relationship between length and width.
Conclusion: You've Got This!
There you have it, guys! We tackled a rectangle area problem. We found the length of the rectangle, identified the relevant equation, and verified our solution. Remember, the key steps are:
- Understand the problem: Identify what's given and what you need to find. What are the relationships between the variables?
- Set up the equation: Translate the word problem into a mathematical equation using variables.
- Solve the equation: Use algebra to isolate the unknown variable (in this case, the width).
- Find the length: Use the value you found for the width to calculate the length.
- Check your answer: Make sure your answer makes sense and satisfies the conditions of the problem. Always double-check. Make sure you didn't make any arithmetic errors.
These steps can be applied to all sorts of area and perimeter problems. Now go forth and conquer those math problems! You're well-equipped to handle similar problems, so keep practicing, and you'll become a math whiz in no time! Keep in mind that practice makes perfect. The more problems you solve, the better you'll become at recognizing patterns and solving them efficiently. Keep going, guys! You got this. Good luck!