Answer
When all three machines are running, the intensity level is $95.2~dB$
This intensity level is only $2.2~dB$ higher than the intensity level of the loudest machine.
Work Step by Step
We can find the intensity of the 85-dB machine:
$\beta = 10~log\frac{I}{I_0}$
$85 = 10~log\frac{I}{I_0}$
$8.5 = log\frac{I}{I_0}$
$10^{8.5} = \frac{I}{I_0}$
$I = (10^{8.5})~I_0$
$I = (10^{8.5})~(1.0\times 10^{-12}~W/m^2)$
$I = 3.16\times 10^{-4}~W/m^2$
We can find the intensity of the 90-dB machine:
$\beta = 10~log\frac{I}{I_0}$
$90 = 10~log\frac{I}{I_0}$
$9.0 = log\frac{I}{I_0}$
$10^{9.0} = \frac{I}{I_0}$
$I = (10^{9.0})~I_0$
$I = (10^{9.0})~(1.0\times 10^{-12}~W/m^2)$
$I = 1.0\times 10^{-3}~W/m^2$
We can find the intensity of the 93-dB machine:
$\beta = 10~log\frac{I}{I_0}$
$93 = 10~log\frac{I}{I_0}$
$9.3 = log\frac{I}{I_0}$
$10^{9.3} = \frac{I}{I_0}$
$I = (10^{9.3})~I_0$
$I = (10^{9.3})~(1.0\times 10^{-12}~W/m^2)$
$I = 2.0\times 10^{-3}~W/m^2$
We can find the total intensity from all three machines:
$I = (3.16\times 10^{-4}~W/m^2)+(1.0\times 10^{-3}~W/m^2)+(2.0\times 10^{-3}~W/m^2)$
$I = 3.316\times 10^{-3}~W/m^2$
We can find the intensity level:
$\beta = 10~log\frac{I}{I_0}$
$\beta = 10~log\frac{3.316\times 10^{-3}~W/m^2}{1.0\times 10^{-12}~W/m^2}$
$\beta = 95.2~dB$
When all three machines are running, the intensity level is $95.2~dB$
This intensity level is only $2.2~dB$ higher than the intensity level of the loudest machine.
