Suppose a speaker is sending out a signal of a given power uniformly in all dire

Question

Suppose a speaker is sending out a signal of a given power uniformly in all directions. You are located a certain distance from the speaker measuring the intensity of the sound. Given the following different situations, rank the measured intensity (1 = the lowest intensity).

Power of speaker is 9P and distance to speaker is (1/3)d.

Power of speaker is (1/4)P and distance to speaker is (1/2)d.

Power of speaker is 4P and distance to speaker is 2d.

Power of speaker is P and distance to speaker is d.

Power of speaker is 3P and distance to speaker is 3d.

Explanation / Answer

The sound intensity decreases with distance to source. Intensity and distance can be expressed as:
I = Pi/4*?*r^2
where:

1) Pi = sound power (W)
2) ? = 3.14
3) r = radius or distance from source (m)
Now you just insert values, to calculate I

(1) 9P and distance to speaker is (1/3)d.
Pi=9P r=1/3d
I=9P/(4*pi*(0.333d)^2)=20.29 *P/(pi*d^2) W/m^2

(2)Power of speaker is (1/4)P and distance to speaker is (1/2)d.

Pi=0.25P r=0.5d
I=0.25P/(4*pi*(0.5d)^2)=0.25 *P/(pi*d^2) W/m^2

(3)Power of speaker is 4P and distance to speaker is 2d.
Pi=4P r=2d
I=4P/(4*pi*(2d)^2)=0.25 *P/(pi*d^2) W/m^2

(4) Power of speaker is P and distance to speaker is d.
Pi=P r=d
I=P/(4*pi*d^2)=0.25 *P/(pi*d^2) W/m^2

(5)Power of speaker is 3P and distance to speaker is 3d.
Pi=3P r=3d
I=3P/(4*pi*(3d)^2)= 0.0833 *P/(pi*d^2) W/m^2

Now if you arrange them in increasing order, you get:

(5)<(2)=(3)=(4) <(1)
Therefore
(5) is 1
(2),(3),(4) is 2
(1) are 3

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