In circle $S$ with $m \angle R S T=112^{\circ}$ and $R S=7$, find the area of sector RST. Round to the nearest hundredth.

Step 1 ：Find the base angle of triangle RST: \(\frac{180 - 112}{2} = 34^\circ\)

Step 2 ：Use the Law of Sines to find the radius: \(\frac{7}{\sin{34^\circ}} = 6.259\)

Step 3 ：Calculate the area of sector RST: \(\frac{112}{360} \times \pi \times 6.259^2 = 38.29\)

Step 4 ：\(\boxed{38.29}\)