Answer
(a) $P = 785~W$
(b) The sound level would be 85 dB at a distance of 445 m
Work Step by Step
(a) We can calculate the intensity of the sound wave as:
$\beta = 10~log(\frac{I}{I_0})$
$log(\frac{I}{I_0})= \frac{\beta}{10}$
$10^{log(\frac{I}{I_0})}= 10^{\frac{\beta}{10}}$
$I= I_0~10^{\frac{\beta}{10}}$
$I= (10^{-12}~W/m^2)~10^{\frac{130}{10}}$
$I = 10~W/m^2$
We then find the power output of the speaker;
$P = I~A$
$P = I~(4\pi~R^2)$
$P = (10~W/m^2)(4\pi)(2.5~m)^2$
$P = 785~W$
(b) We can calculate the intensity of the sound wave when the sound level is 85 dB.
$\beta = 10~log(\frac{I}{I_0})$
$log(\frac{I}{I_0})= \frac{\beta}{10}$
$10^{log(\frac{I}{I_0})}= 10^{\frac{\beta}{10}}$
$I= I_0~10^{\frac{\beta}{10}}$
$I= (10^{-12}~W/m^2)~10^{\frac{85}{10}}$
$I = 3.16\times 10^{-4}~W/m^2$
We can find the distance R from the speaker.
$I~A = P$
$I~(4\pi~R^2) = P$
$R^2 = \frac{P}{I~(4\pi)}$
$R = \sqrt{\frac{P}{I~(4\pi)}}$
$R = \sqrt{\frac{785~W}{(3.16\times 10^{-4}~W/m^2)(4\pi)}}$
$R = 445~m$
The sound level would be 85 dB at a distance of 445 m.
