Calculating Sound Power from Intensity and Distance

[shards5]

Homework Statement

A spherical source radiates sound uniformly in all directions. At a distance of 9 m, the sound intensity level is 100 dB. What power is radiated by this source?

Just a simple answer check to see if my answer is reasonable.

Homework Equations

$$\beta$$ = 10 db log (I/I₀)
Intensity = power/area

The Attempt at a Solution

First convert the given decibels to Intensity then convert the found intensity to power via the second equation.
100 dB = 10 dB log (I/(1*10^-12)
Divide by 10 on both sides.
10 = log ((I/(1*10^-12)
Raise both sides to the tenth power to get rid of log
10¹⁰ = (I/(1*10^-12))
Multiple both sides by 1*10^-12 I get. . .
I = 0.01 (Can someone confirm that I doesn't have units?)
Plugging in I to the second equation I get. . .
0.01 = $$\frac{Power}{4*\pi*9^2}$$
Divide both sides by area I get Power. . .
Power = 10.178 (I know the units should be watt but if Intensity has no units and area only gives m² then how do I get the units for watt? Does this mean this is wrong?)

Thanks in advance for your help.

 

[collinsmark]

Hello shards5,

First convert the given decibels to Intensity then convert the found intensity to power via the second equation.
100 dB = 10 dB log (I/(1*10^-12)
Divide by 10 on both sides.
10 = log ((I/(1*10^-12)
Raise both sides to the tenth power to get rid of log
10¹⁰ = (I/(1*10^-12))
Multiple both sides by 1*10^-12 I get. . .
I = 0.01 (Can someone confirm that I doesn't have units?)

It does have units! :tongue: Here, your 10¹⁰ figure does not have units because it is a ratio of intensities in terms of power fluxes (a ratio of things with the same units has no units because the units cancel). But I₀ Has units! That means your I = 0.01 has units too (which you should be able to figure out going forward from here).

The standard reference sound intensity is

I₀ = 1 x 10^-12 W/m²

(i.e. units of Watts per square meter).

 

[shards5]

Ah, then the whole problem makes sense since m² cancels out nicely at the end leaving me with just Watts. Thanks a lot!