A classical mechanics problem by Jason Hu

The maximum torque output from the engine of a new experimental car of mass m is τ . The maximum rotational speed of the engine is ω. The engine is designed to provide a constant power output P. The engine is connected to the wheels via a perfect transmission that can smoothly trade torque for speed with no power loss. The wheels have a radius R, and the coefficient of static friction between the wheels and the road is µ. The maximum sustained speed v the car can drive up a 30 degree incline is of the form kP/100mg, where k is a constant. Find k. Assume no frictional losses and assume µ is large enough so that the tires do not slip.