Angle between curves

The above curves intersect in the first quadrant of the xy-plane at $2\sqrt{x} = e^{-x/2} \Rightarrow x \approx 0.203888$ . The angle between the two curves is just the sum of the angles their respective tangents make with the x-axis. These are found by computing the slopes at their common intersection point:

$\frac{d}{dx} 2\sqrt{x} = \frac{1}{\sqrt{x}} |_{x=0.203888} = 2.21465$ and $\theta_{1} = arctan(2.21465) = 65.7$ degrees

$\frac{d}{dx} e^{-x/2} = -\frac{1}{2} e^{-x/2} |_{x=0.203888} = -0.45154$ and $\theta_{2} = arctan(0.45154) = 24.3$ degrees

Hence, $\theta_{1} + \theta_{2} = 90$ degrees, or $\boxed{\frac{\pi}{2}}$ radians.