Exploring the Area of a Rectangle with Expressions for Length and Width

Understanding the area of a rectangle involves knowing both its length and width. Often, these dimensions are not given as simple numerals but as algebraic expressions. This article explores the area of a rectangle whose length is expressed as (x 5) and its width as (x - 5). Let's delve into the calculations and explore how different values of (x) affect the area.

Calculating the Area

To find the area of the rectangle, we use the formula for the area of a rectangle:

[ text{Area} text{length} times text{width} ]

Given:

[ text{length} x 5 ] [ text{width} x - 5 ]

F.O.I.L Method

Let's use the F.O.I.L. (First, Outer, Inner, Last) method to multiply the length and width expressions:

[ (x 5)(x - 5) x^2 - 5x 5x - 25 ]

Simplifying:

[ (x 5)(x - 5) x^2 - 25 ]

Therefore, the area of the rectangle is:

[ text{Area} x^2 - 25 text{ (units squared)} ]

Considering Different Values of (x)

The area (x^2 - 25) can be factored as:

[ (x 5)(x - 5) ]

This means that for the dimensions to form a valid rectangle (i.e., length > 0 and width > 0), (x) must be greater than 5. If (x) is less than or equal to 5, one of the dimensions (either length or width) would be non-positive, which does not form a valid rectangle.

Example Calculations and Verifications

Let's verify the area for different values of (x):

Example: (x 11)

[ text{Length} x 5 11 5 16 ] [ text{Width} x - 5 11 - 5 6 ] [ text{Area} 16 times 6 96 text{ (units squared)} ]

Example: (x 10)

[ text{Length} x 5 10 5 15 ] [ text{Width} x - 5 10 - 5 5 ] [ text{Area} 15 times 5 75 text{ (units squared)} ]

Example: (x 9)

[ text{Length} x 5 9 5 14 ] [ text{Width} x - 5 9 - 5 4 ] [ text{Area} 14 times 4 56 text{ (units squared)} ]

From the above examples, we can see that as (x) increases, the difference between the length and the width increases, leading to a larger area. Conversely, as (x) decreases, the area decreases.

Conclusion

We have explored the relationship between (x) and the area of a rectangle when the length is (x 5) and the width is (x - 5). The area is a quadratic expression, and it is valid only for (x > 5). Understanding this relationship is fundamental in many mathematical and real-life applications.