Loudness of sound question

1. Aug 23, 2010

Ryker

1. The problem statement, all variables and given/known data
The speaker receives 10W of electrical power from the amplifier, 4% of which it transforms into sound flow, propagating into a half-space (in front of the speaker) indiscriminately in all directions. How loud is the sound 10m from the speaker?

2. Relevant equations
$$L_\mathrm{I}=10\, \log_{10}\left(\frac{I_1}{I_0}\right)\ \mathrm{dB} \,$$
$$I = \frac{W}{S}$$

3. The attempt at a solution
The solution is pretty easy to get if you just plug the data into the equations, taking

$$S = 4\pi r^{2}$$

But that's the thing I don't quite get. If all of the power that the speaker emits is transformed into sound that only propagates in a half-space, why isn't $$S = 2\pi r^{2}$$? I mean, isn't that the point of a speaker, so as to curtail the drawback of sound propagating in all directions and directing its power into only the desired one, thus making it appear louder to our ears?