A rural arterial highway is being designed through rolling terrain with a design speed of $100 \text{ km/h}$. The local highway agency specifies a maximum superelevation rate ($e_{max}$) of $8.0\%$ ($0.08$). According to guidelines, the maximum safe coefficient of side friction ($f_{max}$) for a speed of $100 \text{ km/h}$ is $0.12$. Calculate the absolute minimum radius ($R_{min}$) that can be used for a horizontal curve on this alignment.

An existing horizontal curve has a radius of $600 \text{ meters}$. The highway is being upgraded to a design speed of $110 \text{ km/h}$. If the design coefficient of side friction at this speed is $0.11$, what is the required rate of superelevation ($e$) to safely negotiate this curve? Express the answer as a percentage.

A circular curve with a radius of $300 \text{ meters}$ is being designed for a highway with a speed limit of $80 \text{ km/h}$. To ensure passenger comfort, the rate of change of centrifugal acceleration ($C$) is limited to $0.6 \text{ m/s}^3$. Determine the required minimum length of the transition curve (spiral).

A two-lane highway ($n=2$) is designed with a horizontal curve radius of $200 \text{ m}$ for a speed of $60 \text{ km/h}$. The standard rigid wheelbase length of the design vehicle ($l$) is $6.0 \text{ m}$. Calculate the total extra widening required on this curve.

A two-lane mountain highway features a sharp horizontal curve where the inside edge is flanked by a steep rock cut (a cliff face). The design speed is $80 \text{ km/h}$, and the Stopping Sight Distance (SSD) required is $130 \text{ meters}$. A local politician wants to widen the shoulder and move the road closer to the rock face to save money on constructing a retaining wall on the outside drop-off. As the design engineer, analyze the safety implications of this proposal.