Answer
(a) The eardrum receives $1.58\times 10^{-11}~joules$ of energy per second.
(b) It would take 2010 years to receive 1.0 J of energy.
Work Step by Step
(a) We can calculate the intensity of the sound wave.
$\beta = 10~log(\frac{I}{I_0})$
$log(\frac{I}{I_0})= \frac{\beta}{10}$
$10^{log(\frac{I}{I_0})}= 10^{\frac{\beta}{10}}$
$I= I_0~10^{\frac{\beta}{10}}$
$I= (10^{-12}~W/m^2)~10^{\frac{55}{10}}$
$I = 3.16\times 10^{-7}~W/m^2$
We can find the power received by the eardrum each second.
$P = I~A$
$P = (3.16\times 10^{-7}~W/m^2)(5.0\times 10^{-5}~m^2)$
$P = 1.58\times 10^{-11}~W$
The eardrum receives $1.58\times 10^{-11}~joules$ of energy per second.
(b) We can calculate the time to receive 1.0 J of energy.
$t = \frac{1.0~J}{1.58\times 10^{-11}~J/s}$
$t = 6.33\times 10^{10}~s$
We can convert the time to units of years.
$t = (6.33\times 10^{10}~s)(\frac{1~hr}{3600~s})(\frac{1~day}{24~hr})(\frac{1~yr}{365~days})$
$t = 2010~years$
It would take 2010 years to receive 1.0 J of energy.