To find the maximum angle at which the crowd observes the launch of the rocket, we can use the right triangle formed by the height of the rocket and the distance of the crowd from the launch site.
Let's denote:
- The height of the rocket, \( h = 300\sqrt{3} \) meters.
- The distance from the crowd to the launch site, \( d = 300 \) meters.
We want to find the angle \( \theta \) where the crowd is observing the rocket at its maximum height. The angle \( \theta \) can be calculated using the tangent function:
\[ \tan(\theta) = \frac{h}{d} \]
Substituting the values we have:
\[ \tan(\theta) = \frac{300\sqrt{3}}{300} \]
This simplifies to:
\[ \tan(\theta) = \sqrt{3} \]
To find the angle \( \theta \), we take the arctangent:
\[ \theta = \tan^{-1}(\sqrt{3}) \]
We know that \( \tan(60^\circ) = \sqrt{3} \). Therefore:
\[ \theta = 60^\circ \]
Thus, the maximum angle at which the crowd observes the launch is:
\[ \boxed{60^\circ} \]