Answer:
Let's denote the sides of the triangle as \(11x\), \(16x\), and \(24x\) (where \(x\) is a scaling factor).
Given that the perimeter of the triangle is \(510\) feet, we can write the equation:
\[11x + 16x + 24x = 510\]
Solve for \(x\):
\[51x = 510\]
\[x = 10\]
Now we know the lengths of the sides are \(110\), \(160\), and \(240\) feet.
To calculate the area of the triangle using Heron's formula:
\[s = \frac{a + b + c}{2} = \frac{110 + 160 + 240}{2} = 255\]
Area (\(A\)) = \(\sqrt{s(s - a)(s - b)(s - c)}\)
\[A = \sqrt{255 \cdot (255 - 110) \cdot (255 - 160) \cdot (255 - 240)}\]
\[A = \sqrt{255 \cdot 145 \cdot 95 \cdot 15}\]
\[A = \sqrt{814875375}\]
\[A \approx 28,566.38\]
So, the area of the triangle is closest to \(\text{A. 7,258.74 ft²}\).