(i) To find the speed of the particle, we differentiate the displacement \(x\) with respect to time \(t\). The velocity \(\dot{x}\) is given by:

\(\dot{x} = \frac{d}{dt} \left( \frac{1}{2}t^2 + \frac{1}{30}t^3 \right) = t + \frac{1}{10}t^2\)

Substitute \(t = 10\) into the velocity equation:

\(\dot{x} = 10 + \frac{1}{10}(10)^2 = 10 + 10 = 20 \text{ m/s}\)

(ii) To find the acceleration, differentiate the velocity \(\dot{x}\):

\(\ddot{x} = \frac{d}{dt}(t + \frac{1}{10}t^2) = 1 + \frac{1}{5}t\)

The initial acceleration \(\ddot{x}(0) = 1\).

We need \(\ddot{x}(t) = 2 \times \ddot{x}(0)\), so:

\(1 + \frac{1}{5}t = 2\)

Solving for \(t\):

\(\frac{1}{5}t = 1\)

\(t = 5\)