The formula is 4fx=y2, where f is the focus, x is the depth and y the fixed radius. This applies if the reflector is cylindrical (constant parabolic cross-section) or if its a paraboloid (rotated parabola).
y=38/2=19 in, x=4 in, so 16f=192=361, f=361/16=22.5625in.
The equation of the parabola is x=4y2/361.
The red curve represents the parabola of the reflector. The green line (length 38") represents the limit of the parabola (depth 4"), its diameter. The focal point is shown, where all incident light would be focused.