Intensity, Radius (Calculations)

[Gunman]

Homework Statement

A point source of sound emits energy equally in all directions at a constant rate and a person 8m from the source listens. After a while, the intensity of the source is halved. If the person wishes the sound to be seem as loud as before, how far should he be now?

Homework Equations

intensity is inversely proportional to radius(distance from the source)

The Attempt at a Solution

I = (K)/(r^2)
I*(r^2)=K
So i presumed that since K has to be constant, and I has been halved. The radius would be (2)^(1/2). So why isn't the answer (2^(1/2))r ? I can understand that the answer is not correct because it is not possible for the intensity to be the same when the radius has increased. So can someone point out to me which part I went wrong? Thank you.

 

[Mindscrape]

Since they are inversely proportional the person needs to get closer by some factor. If it were simply

$$I \alpha \frac{K}{r}$$

and the intensity dropped by a factor of two then that is the equivalent of saying that the person went twice as far away, so in the case of the intensity of source being independent then to keep the same sound the person would half to cancel it out by going r/2.

Now in the case of it being inversely proportional to the square of the radius, if the intensity decreases by a half, it could have happened because someone traveled 2^(1/2) away, so if it were actually the source that decreased by a half then the person can cancel that effect just as if they could have caused it by doing the opposite, and traveling 2^(1/2) closer (i.e. r/(2^1/2)).

If it helps to see the algebra in addition to the qualitative observations here is how it would go for the two:
$$I \alpha \frac{1}{2} \frac{K}{r/2} = \frac{k}{r}$$

$$I \alpha \frac{1}{2} \frac{K}{(r/\sqrt{2})^2} = \frac{k}{r^2}$$