Como Encontrar O Valor De X Em Um Retângulo Com Lados (x - 4) Cm E (x + 4) Cm E Área De 48 Cm²
Introduction
In geometry, rectangles hold a fundamental place, and understanding their properties is crucial. One such property is the relationship between the sides and the area of a rectangle. This article aims to delve into a specific problem involving a rectangle, where the lengths of its sides are expressed in terms of an unknown variable, x, and the area is given. Our goal is to determine the value of x that satisfies these conditions. We will explore the concept of area, the formula for calculating the area of a rectangle, and how to apply algebraic techniques to solve for the unknown. This problem not only reinforces geometrical concepts but also hones problem-solving skills in algebra. The application of these skills extends beyond the classroom, finding relevance in real-world scenarios such as construction, design, and spatial planning. Let's embark on this journey to solve for x and unravel the mystery behind this rectangular puzzle.
Problem Statement
The problem presents us with a rectangle whose sides are expressed algebraically. Specifically, the lengths of the sides are given as (x - 4) cm and (x + 4) cm. These expressions indicate that the side lengths are dependent on the value of x. The area of the rectangle is provided as 48 cm². Our task is to find the value of x that makes these conditions true. This problem is a classic example of how algebra and geometry intertwine, requiring us to apply both geometric principles and algebraic manipulation to arrive at the solution. The algebraic expressions for the sides introduce a layer of complexity, as we need to work with variables and equations rather than simple numerical values. This makes the problem more engaging and necessitates a systematic approach to solve it.
Understanding the Area of a Rectangle
The area of a rectangle is a fundamental concept in geometry, representing the amount of two-dimensional space enclosed within the rectangle. It is a measure of the surface covered by the rectangle and is typically expressed in square units, such as square centimeters (cm²) in this case. The area of a rectangle is calculated by multiplying its length by its width. This simple yet powerful formula forms the basis for many geometric calculations and is essential in various applications, from calculating the size of a room to determining the amount of material needed for a project. Understanding the concept of area is crucial for solving problems related to rectangles and other geometric shapes. It allows us to relate the dimensions of a shape to its spatial extent, providing a quantitative measure of its size. In this problem, the area is given, and we need to work backward to find the dimensions, which adds an interesting twist to the standard area calculation.
Setting up the Equation
To solve for x, we need to translate the given information into a mathematical equation. The area of a rectangle is given by the formula:
Area = Length × Width
In our problem, the lengths of the sides are (x - 4) cm and (x + 4) cm, and the area is 48 cm². Substituting these values into the formula, we get:
48 = (x - 4) × (x + 4)
This equation is a quadratic equation, which means it involves a variable raised to the power of two. Quadratic equations are common in many areas of mathematics and physics and often arise in geometric problems like this one. The equation we have set up is a crucial step in solving for x, as it mathematically represents the relationship between the sides and the area of the rectangle. Solving this equation will give us the value(s) of x that satisfy the given conditions. The next step involves simplifying and solving this quadratic equation.
Solving the Quadratic Equation
Now that we have the equation:
48 = (x - 4) × (x + 4)
We need to simplify and solve it for x. The expression (x - 4) × (x + 4) is a special case known as the difference of squares. It can be simplified using the formula:
(a - b) × (a + b) = a² - b²
Applying this formula to our equation, we get:
48 = x² - 16
Now, we have a simpler quadratic equation. To solve for x, we need to isolate x². We can do this by adding 16 to both sides of the equation:
48 + 16 = x²
64 = x²
To find x, we take the square root of both sides:
x = ±√64
x = ±8
This gives us two possible values for x: 8 and -8. However, in the context of this problem, x represents a length, which cannot be negative. Therefore, we discard the negative solution.
Considering the Context: Side Lengths
We have found two possible values for x: 8 and -8. However, it is crucial to consider the context of the problem. The side lengths of the rectangle are given as (x - 4) cm and (x + 4) cm. Since lengths cannot be negative, we need to check if both values of x make sense in this context.
If we substitute x = -8 into the expressions for the side lengths, we get:
Side 1 = (-8) - 4 = -12 cm
Side 2 = (-8) + 4 = -4 cm
Both side lengths are negative, which is not physically possible. Therefore, x = -8 is not a valid solution in this context.
Now, let's substitute x = 8 into the expressions for the side lengths:
Side 1 = (8) - 4 = 4 cm
Side 2 = (8) + 4 = 12 cm
Both side lengths are positive, which is physically possible. Therefore, x = 8 is a valid solution.
Final Answer: The Value of x
After solving the quadratic equation and considering the context of the problem, we have determined that the value of x is 8. This value makes the side lengths of the rectangle positive and consistent with the given area of 48 cm². Therefore, the final answer is:
x = 8
This solution demonstrates the importance of not only solving equations correctly but also interpreting the solutions in the context of the problem. In this case, the negative solution was mathematically valid but physically impossible, highlighting the need for careful consideration of the problem's constraints.
Conclusion
In this article, we successfully solved a geometric problem involving a rectangle, where the lengths of its sides were expressed in terms of an unknown variable, x, and the area was given. We started by understanding the problem statement and the concept of the area of a rectangle. We then set up a quadratic equation based on the given information and solved it using algebraic techniques. We obtained two possible values for x, but we had to consider the context of the problem to determine the valid solution. We realized that side lengths cannot be negative, so we discarded the negative solution and concluded that x = 8 is the correct answer. This problem showcased the interplay between algebra and geometry, requiring us to apply both geometric principles and algebraic manipulation. It also emphasized the importance of interpreting solutions in the context of the problem, a crucial skill in mathematical problem-solving. The ability to solve such problems is valuable in various fields, including engineering, architecture, and design, where geometric and spatial reasoning are essential.
