To find the area of a rectangle, you use the formula:
\[ \text{Area} = \text{length} \times \text{width} \]
In Natalia's case, the dimensions of the rug are 57 inches and 41 inches. Thus, the area of the rug can be calculated using:
\[ A = 57 \times 41 \]
Now, let's evaluate the options you provided:
A. \(57^2 - 41^2\) - This expression represents the difference of squares and does not relate to the area of the rectangle.
B. \(57^2 + 41^2\) - This expression calculates the sum of the squares of the lengths, which also does not give the area.
C. \(49^2 - 8^2\) - This expression can be rewritten using the difference of squares formula \(a^2 - b^2 = (a-b)(a+b)\). Here, \(49^2 - 8^2\) simplifies to \((49 - 8)(49 + 8) = 41 \times 57\), which indeed equals the area of the rug.
D. \(49^2 + 8^2\) - Similar to option B, this is the sum of squares and doesn't relate to the area.
Based on this analysis, the correct expression that Natalia can use to find the area of the rug is:
C. \(49^2 - 8^2\).