Rectangle Area: Finding The Correct Statement

Let's dive into a fun geometry problem involving rectangles, areas, and a bit of algebra! We're given a rectangle with a specific area and width, and our mission is to determine which statement about this rectangle is true. Think of it as a detective game where we need to uncover the correct clue. This problem not only tests our understanding of geometric formulas but also our ability to manipulate algebraic expressions. We'll be using concepts like factoring quadratic expressions and calculating perimeters. So, grab your thinking caps, and let's get started on this mathematical adventure! Understanding the relationship between area, width, and length is key here, and we'll break down each step to make sure it's crystal clear. Remember, mathematics is like a puzzle, and each piece fits perfectly to reveal the solution. Let’s explore this intriguing problem together and unlock the mystery of this rectangle!

Breaking Down the Problem

Before we jump into solving, let's make sure we fully understand the information we have. The problem tells us we have a rectangle. A rectangle, as we know, is a four-sided shape with opposite sides that are equal in length and four right angles. The area of this rectangle is given by the expression $4x^2 + 39x - 10$ square units. Remember, the area of a rectangle is calculated by multiplying its length and width. We're also given that the width of the rectangle is $(x + 10)$ units. Now, our goal is to figure out which of the provided statements is true about this rectangle. The statements involve things like whether the rectangle is a square, its length, and its perimeter. To solve this, we'll need to use our knowledge of algebra to find the length of the rectangle and then use that information to evaluate the given statements. This is a classic problem-solving scenario where we take what we know, apply some mathematical tools, and arrive at a conclusion. So, let's start by figuring out how to find the length of this rectangle!

Finding the Length

Okay, so we know the area of a rectangle is found by multiplying its length and width. Mathematically, we can write this as: Area = Length × Width. In our case, we know the area ($4x^2 + 39x - 10$) and the width ($x + 10$). What we need to find is the length. To do this, we can rearrange the formula to solve for the length: Length = Area / Width. Now, we can substitute the given expressions into this formula: Length = $(4x^2 + 39x - 10) / (x + 10)$. This looks like an algebraic fraction, and to simplify it, we'll need to factor the quadratic expression in the numerator. Factoring is like the reverse of expanding brackets, and it helps us break down complex expressions into simpler ones. In this case, we need to factor $4x^2 + 39x - 10$. Factoring this quadratic expression is a crucial step, and there are different techniques we can use, such as trial and error or using the quadratic formula. Once we've factored the numerator, we can see if there are any common factors with the denominator that we can cancel out. This will give us a simplified expression for the length of the rectangle.

Factoring the Quadratic Expression

The expression we need to factor is $4x^2 + 39x - 10$. Factoring a quadratic expression like this involves finding two binomials (expressions with two terms) that, when multiplied together, give us the original quadratic. There are several methods to do this, but one common approach is to look for two numbers that multiply to the product of the leading coefficient (4) and the constant term (-10), which is -40, and add up to the middle coefficient (39). In this case, the two numbers are 40 and -1. Now, we can rewrite the middle term (39x) using these two numbers: $4x^2 + 40x - x - 10$. Next, we group the terms in pairs: $(4x^2 + 40x) + (-x - 10)$. We can now factor out the greatest common factor (GCF) from each pair. From the first pair, we can factor out 4x, and from the second pair, we can factor out -1: $4x(x + 10) - 1(x + 10)$. Notice that we now have a common factor of $(x + 10)$ in both terms. We can factor this out: $(x + 10)(4x - 1)$. So, the factored form of $4x^2 + 39x - 10$ is $(x + 10)(4x - 1)$. This is a key step in solving our problem!

Simplifying and Finding the Length

Now that we've factored the quadratic expression, we can go back to our equation for the length: Length = $(4x^2 + 39x - 10) / (x + 10)$. We found that $4x^2 + 39x - 10$ can be factored as $(x + 10)(4x - 1)$. So, we can substitute this into the equation: Length = $((x + 10)(4x - 1)) / (x + 10)$. Now we have a common factor of $(x + 10)$ in both the numerator and the denominator. We can cancel this common factor out, which simplifies the expression: Length = $4x - 1$. So, we've successfully found the length of the rectangle! It's $4x - 1$ units. This was a crucial step because now we can use this information to evaluate the statements given in the problem. Remember, the statements involve whether the rectangle is a square, its length, and its perimeter. We've already found the length, so let's move on to evaluating the statements.

Evaluating the Statements

Now that we know the length of the rectangle is $(4x - 1)$ units and the width is $(x + 10)$ units, we can evaluate the given statements to see which one is true. Let's look at the statements one by one:

• Statement A: The rectangle is a square.

  For a rectangle to be a square, all its sides must be equal. This means the length and the width must be the same. So, we need to check if $(4x - 1)$ is equal to $(x + 10)$. To do this, we can set up the equation $4x - 1 = x + 10$ and solve for x. Subtracting x from both sides gives $3x - 1 = 10$. Adding 1 to both sides gives $3x = 11$. Dividing both sides by 3 gives $x = 11/3$. So, the length and width are only equal when $x = 11/3$. However, this doesn't mean the rectangle is always a square. It's only a square for this specific value of x. Therefore, statement A is not generally true.

• Statement B: The rectangle has a length of $(2x - 5)$ units.

  We already found that the length of the rectangle is $(4x - 1)$ units. This is clearly different from $(2x - 5)$ units. So, statement B is false.

• Statement C: The perimeter of the rectangle is $(10x + 18)$ units.

  The perimeter of a rectangle is the sum of the lengths of all its sides. Since a rectangle has two lengths and two widths, the perimeter is given by the formula: Perimeter = 2 × (Length + Width). We know the length is $(4x - 1)$ and the width is $(x + 10)$. Substituting these into the formula, we get: Perimeter = 2 × $((4x - 1) + (x + 10))$. Now, we simplify the expression inside the parentheses: Perimeter = 2 × $(5x + 9)$. Finally, we distribute the 2: Perimeter = $10x + 18$. This matches statement C! So, statement C is true.

Conclusion

After carefully analyzing the problem, factoring the quadratic expression, finding the length, and evaluating each statement, we've reached our conclusion. Statement C: The perimeter of the rectangle is $(10x + 18)$ units is the true statement. This problem was a great exercise in applying our knowledge of algebra and geometry. We used factoring, simplifying expressions, and the formulas for area and perimeter to solve it. Remember, practice is key to mastering these concepts. The more problems you solve, the more comfortable you'll become with different techniques and approaches. Keep exploring, keep learning, and most importantly, have fun with mathematics! If you're keen to explore more about rectangles and their properties, a great resource to check out is Khan Academy's Geometry section on Rectangles. It offers a wealth of information and practice problems to further enhance your understanding.