How to Calculate the Area of a Rectangle: Step-by-Step Guide
Quick Answer
To calculate the area of a rectangle, use the formula A = length × width. In this case, the area function is A(x) = x(x - 4) = x² - 4x, where the width is reduced by 4 feet.
Calculating the area of a rectangle is a fundamental concept in geometry that can be easily mastered. In this scenario, we have a rectangular deck where the dimensions are defined based on the variable x. Understanding how to derive the area function from these dimensions is essential for solving related problems.
The first step is to identify the dimensions of the rectangle. According to the problem, the top and bottom sides of the deck have a length of x feet each. This means the length of the rectangle is represented as:
- **Length = x**
Next, we look at the left and right sides of the rectangle. Each of these sides uses a beam with 4 feet cut off, which means that the width of the rectangle is:
- **Width = x - 4**
Now that we have the length and width, we can calculate the area of the rectangle. The formula for the area (A) of a rectangle is:
- **Area = Length × Width**
Substituting the expressions we have for length and width into this formula gives us:
- **A(x) = x × (x - 4)**
Expanding this expression, we get:
- **A(x) = x² - 4x**
This means the area function is A(x) = x² - 4x. If you had chosen A(x) = x² - 4, that would be incorrect because that expression comes from multiplying (x + 2)(x - 2), not the correct dimensions of the rectangle.
Understanding why the width is x - 4 instead of just 4 is crucial. The width is not a fixed number but a variable dependent on x. The problem states that the beams are cut down from the original width, which is why it is expressed as x - 4, indicating that the original width is reduced by 4 feet.
This concept can be applied in real-world situations, such as constructing decks or other rectangular structures. By understanding how to manipulate dimensions and calculate areas, you can make accurate measurements and estimations for building projects.
In summary, always remember to identify your dimensions carefully and apply the area formula correctly to ensure accurate calculations. With practice, these concepts will become second nature, and you'll be able to tackle more complex geometric problems with confidence.
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