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Have you ever wondered how to calculate the dimensions of a rectangle when you only know its perimeter and area? In this article, we'll dive into the mathematical secrets behind a rectangle's geometry. Let's begin by setting the stage with a intriguing question: Can you determine the length and width of a rectangle if you know its perimeter is 17x units and its area is 15x square units?
To solve this mystery, we first need to establish two equations, one for the perimeter and one for the area. Let's tackle the perimeter equation first. What does the perimeter represent? It's the sum of all the sides of the rectangle. Imagine you have a rectangle with sides measuring 4x + 2 units and 2.5 units. How would you express the perimeter in an equation?
Pause for a moment and try to write that equation yourself. Ready? Here's the catch: the perimeter equals 17x units. So, how do we represent this mathematically? The equation would be:
[ 2 \times (4x + 2) + 2 \times 2.5 = 17x ]
But wait, there's more! What about the area? The area of a rectangle is calculated by multiplying its base by its height. Now, considering our rectangle has a base of 2.5 units and a height of 4x + 2 units, how would you express the area in an equation? The area is given as 15x square units. So, our area equation becomes:
[ 2.5 \times (4x + 2) = 15x ]
Or, if you prefer, you can flip the terms:
[ 15x = 2.5 \times (4x + 2) ]
Now that we have our equations, let's take a step back and consider the implications. These equations hold the key to unlocking the dimensions of our rectangle. But how do we solve them? And what does it mean for the rectangle's shape and size?
Stay tuned as we delve deeper into the algebraic manipulations needed to solve these equations. But before we do, let's ponder this: How do these equations reflect the relationship between the perimeter and area of a rectangle? What insights can we gain from understanding this relationship?
In our next segment, we'll explore the solutions to these equations and what they reveal about the rectangle's dimensions. So, keep reading and let's unravel the geometry of this rectangle together!
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