Answer
$78.93\ ft^2$.
Work Step by Step
1. Given $\theta=50^\circ=\frac{50}{180}\pi=\frac{5}{18}\pi$, with a rim of $s=25\ ft$, we have the radius $r=\frac{s}{\theta}=\frac{25}{\frac{5}{18}\pi}=\frac{90}{\pi}\ ft$.
2. Add a 3 foot wide walk will give a new radius $R=r+3$ and the area of the paving blacks is
$A=\frac{1}{2}R^2\theta-\frac{1}{2}r^2\theta\\=\frac{1}{2}\theta(R^2-r^2)\\=\frac{1}{2}\theta(R-r)(R+r)\\=\frac{3}{2}\theta(2r+3)\\=\frac{3}{2}(\frac{5}{18}\pi)(\frac{180}{\pi}+3)\\ \approx78.93\ ft^2$.
