You know that the two lines intersect at 710 to each other. The center of the radius of the curve has to lie on the bisector of these two lines, so the angle from the bisector to each line will be 710/2=35.50. The tangent to the curve is 320ft from the intersection of the 2 lines. The line from the center of the curve to the tangent line of the track is also the radius of the curve. We can now make a right triangle with the 320ft length side, the radius of the curve and the bisector with the right angle between the 320ft side and the radius. Using trig we can solve for the radius, r:
r ≅ 228.25ft
To find the arc length, we need to solve for S=(x/360)2πr, where x is the angle between the 2 tangents. This measurement is found by knowing that the two perpendicular radii formed by the tangents plus the angle between the two tangents plus the angle between the two lines must equal 3600. That is:
This gives S=(109/360)2π(228.25) ≅ 434.23ft. This is a lot easier to see if you draw a sketch.