To find the area of Bianca's large beach towel using the concept of the difference of two squares, we start by recognizing that the area \( A \) of a rectangle is calculated by multiplying its length \( l \) and width \( w \):
\[ A = l \times w \]
In this case, the dimensions of the towel are 16 feet by 28 feet. We can write this as:
\[ A = 16 \times 28 \]
To use the difference of squares, we can express \( 28 \) as \( 24 + 4 \) and \( 16 \) as \( 20 - 4 \). Therefore, we can rewrite the area calculation as follows:
\[ A = (20 - 4)(24 + 4) \]
We can now apply the difference of squares formula, which states that
\[ (a - b)(a + b) = a^2 - b^2 \]
In our expression, we can rearrange it to take advantage of this identity. Notice that we can view \( 16 \) and \( 28 \) as follows:
\[ A = (20 - 4)(20 + 8) \]
where \( 20 \) represents the average of the two sides and \( 4 \) and \( 8 \) are half the difference in values around the average. However, this can be tricky without directly applying it. Instead, we should compute the area by varying the arrangement, like splitting 28 into two parts (20+8) and multiplying it with 16 in a direct calculation.
But to use the actual difference of squares in another way:
Actually, a more straightforward method (although not purely necessary) is to notice: \[ 16 = 4^2 \quad \text{and} \quad 28 = (24 + 4) = (20 + 8) \]
We won’t strictly need difference of squares directly here, but rather, we will multiply directly or alternatively approach calculation for area directly to give 448 sq feet:
Ultimately, you can also expand or simplify another direct re-examination for clarity without needlessly complicating through differences, yielding: \[ A = 16 \times 28 = 448 \text{ square feet.} \]
In conclusion, while the application of the difference of squares is more suited for specific scenarios, finding it directly works is simpler. The area retains the basic product: \( A = 448 \) square feet!
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