(i) To find when P returns to O, set \(x = 0\):

\(0.08t^2 - 0.0002t^3 = 0\)

\(t^2(0.08 - 0.0002t) = 0\)

\(t = 0\) or \(0.08 - 0.0002t = 0\)

\(0.0002t = 0.08\)

\(t = \frac{0.08}{0.0002} = 400\) s

To find the speed at O on return, differentiate \(x\) to get velocity \(v\):

\(v = \frac{dx}{dt} = 0.16t - 0.0006t^2\)

At \(t = 400\):

\(v = 0.16 \times 400 - 0.0006 \times 400^2\)

\(v = 64 - 96 = -32\) m/s

Speed is 32 m/s.

(ii)(a) To find the total distance travelled, find the time to the furthest point:

\(v = 0.16t - 0.0006t^2 = 0\)

\(t(0.16 - 0.0006t) = 0\)

\(t = 0\) or \(t = \frac{0.16}{0.0006} \approx 266.67\) s

Distance to furthest point:

\(x = 0.08(266.67)^2 - 0.0002(266.67)^3\)

\(x \approx 1895\) m

Total distance = \(2 \times 1895 = 3790\) m

(ii)(b) Average speed = \(\frac{\text{total distance}}{\text{total time}} = \frac{3790}{400} = 9.48\) m/s