The following is a question from an old UK A-Level Applied Mathematics textbook. I have tried to solve this and agree with the book answer for the first part of the question, but for the second part of the question I get a different answer to the book. Could someone please help me understand what I have done wrong, or whether the book answer is incorrect.

A car of mass 1500 kg has a maximum speed of 150 km h^{-1}on a level road when the engine is exerting its maximum power of 200 kW. Find the resistance to motion at this speed. If this resistance is proportional to the speed of the car, find the maximum speed of the car up a road inclined at arcsin(0.1) to the horizontal.

This is my answer to the question:

LetFbe the force acting on the car andRbe the resistance to motion.

At the maximum speed there is no acceleration; therefore, the forces acting on the car are in equilibrium. ThereforeF=R. We are given

P= 200 x 10^{3}W,m= 1500 kg,v= 150 km h^{-1}= 125/3 m s^{-1}

P=Fv,thereforeR= 4 800 N

Now for the second part:

When travelling at any speedv,R=kv, sok=R/v =4800 / (125/3) = 576/5

When travelling up the hill at maximum speedV

F=P/V= 200 x 10^{3}/VandR= 576V/5

There is no acceleration, so ifFis the force acting on the car

F- 1500g (0.1) =R

200 x 10^{3}/V- 1470 = 576V/5

576V^{2}/5 + 1470V- 200 000 = 0

so

The book gives the answer as 13.3 m s^{-1}. What am I doing wrong?

Many thanks