(a) To find the initial speed u, use the equation for maximum height:

\(0 = u^2 - 2 \times 10 \times 20\)

Solving for \(u\), we get:

\(u^2 = 400\)

\(u = 20 \text{ m/s}\)

(b) To find the total time for which P is at least 15 m above the ground, use the equation:

\(15 = 20t - \frac{1}{2} \times 10 \times t^2\)

Rearranging gives:

\(5t^2 - 20t + 15 = 0\)

Solving this quadratic equation, we find:

\(t = 1 \text{ or } t = 3\)

The total time is \(3 - 1 = 2 \text{ s}\).