We can use conservation of energy to solve this problem. At the maximum height, all of the initial kinetic energy of the skier will be converted into potential energy. We can set the initial kinetic energy equal to the potential energy at the maximum height:
(1/2) * m * v_i^2 = m * g * h_max
where m is the mass of the skier, v_i is the initial velocity, g is the acceleration due to gravity (9.81 m/s^2), and h_max is the maximum height.
Simplifying this equation, we can solve for v_i:
v_i = sqrt(2 * g * h_max)
Plugging in the given values, we get:
v_i = sqrt(2 * 9.81 m/s^2 * 4.5 m) ≈ 9.9 m/s
Therefore, the skier is traveling at a speed of approximately 9.9 m/s at his maximum height of 4.5 meters above the level at the end of the ski jump. Note that the angle theta is not needed to solve this problem, since the maximum height only depends on the initial speed and the vertical distance traveled.