4 A car of mass 800 kg experiences a resistance of magnitude \(k v ^ { 2 } \mathrm {~N}\), where \(k\) is a constant and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the car's speed. The car's engine is working at a constant rate of \(P \mathrm {~W}\). At an instant when the car is travelling on a horizontal road with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(0.75 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At an instant when the car is ascending a hill of constant slope \(12 ^ { \circ }\) to the horizontal with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Show that \(k = 0.900\), correct to 3 decimal places, and find \(P\). The power is increased to \(1.5 P \mathrm {~W}\).
2. Calculate the maximum steady speed of the car on a horizontal road.