A patio is in the shape of a sector. The patio has a central angle of 240° and contains 76.0 square feet of surface area. What is the radius of the patio rounded to 2 decimal places?
The formula for the area of a sector is given by
\[A = \frac{n}{360} \pi r^2\]
where $A$ is the area of the sector, $n$ is the central angle in degrees, and $r$ is the radius of the sector.
We are given that $A = 76.0$ square feet and $n = 240^\circ$. Plugging these values into the formula, we have
\[76.0 = \frac{240}{360} \pi r^2\]
Simplifying this equation, we get
\[r^2 = \frac{76}{\frac{2}{3} \pi} = \frac{3}{2} \cdot \frac{76}{\pi} = \frac{114}{\pi}\]
Taking the square root of both sides, we find
\[r = \sqrt{\frac{114}{\pi}} \approx \boxed{6.00}\]