Explanation:

To find the spiral angle, we first need to understand the concepts of the radius of the circular curve, the length of the transition curve, and the spiral angle.

Radius of the Circular Curve:
The radius of the circular curve refers to the radius of the arc that makes up the curved section of a road or railway track. It is denoted by 'R'.

Length of the Transition Curve:
The transition curve is a curve that connects a straight section of a road or railway track to a circular curve. It is used to gradually change the alignment and increase the comfort of the transition from a straight path to a curved path. The length of the transition curve is denoted by 'L'.

Spiral Angle:
The spiral angle, denoted by 'δ', is the angle subtended by the transition curve at the center of the circular curve.

To find the spiral angle, we are given that the radius of the circular curve is five times the length of the transition curve. Mathematically, this can be represented as:

R = 5L

Deriving the Formula:

To find the spiral angle, we need to determine the relationship between the radius of the circular curve, the length of the transition curve, and the spiral angle.

Consider a small section of the transition curve with an arc length 'ds'. This small section can be approximated as a straight line when the length 'ds' is very small.

Using basic trigonometry, we can find the angle subtended by this small section at the center of the circular curve. The angle 'dδ' can be given as:

dδ = ds / R

Since the length of the transition curve is L, we can express the total spiral angle 'δ' as the sum of all these small angles:

δ = ∫(ds / R)

Substituting R = 5L, we get:

δ = ∫(ds / 5L)

Simplifying the integral, we get:

δ = 1 / 5 ∫(ds / L)

δ = 1 / 5 (L / L)

δ = 1 / 5

Hence, the spiral angle is given by 1/10 radian, which corresponds to option (b).