Answer:
[tex]5.87*10^{-4}W[/tex]
Explanation:
Given that:
[tex]\beta = 89,6 dB[/tex]
[tex]D = 13.0 cm[/tex]
[tex]r = \frac{D}{2}\\ = \frac{13.0}{2}\\ = 6.5 cm\\=0.065m[/tex]
Efficiency = 2.06 % = 0.0206
The intensity level of sound is given by the formula:
[tex]\beta = (10dB) log (\frac{I}{I_o} )[/tex]
[tex]\frac{\beta}{(10dB) } =log (\frac{I}{I_o} )[/tex]
Taking their exponential; we have :
[tex]10^{\frac{\beta}{10dB}}= \frac{I}{I_o}[/tex]
[tex]I = I_o(10^{\frac{\beta}{10dB}})[/tex]
Replacing our values; we have:
[tex]I = (10^{-12}W/m^2)(10^{\frac{89.6}{10dB}})[/tex]
[tex]= 9.12 *10^{-4}W/m^2[/tex]
Power Output;
[tex]P_{out} = IA\\P_{out}=I(\pi r^2)\\P_{out}=(9.12*10^{-4}W/m^2)(3.14)(0.065m)^2\\P_{out}=1.2099048*10^{-5} W[/tex]
The power output that is required to produce a 89.6 dB sound intensity level for a 13.0 cm diameter speaker that has an efficiency of 2.06% is;
[tex]P_{in}= \frac{P_{out}}{0.0206}[/tex]
[tex]P_{in}= \frac{{1.2099048*10^{-5}}}{0.0206}[/tex]
[tex]P_{in}= 5.87*10^{-4}W[/tex]
Therefore, The power output that is required to produce a 89.6 dB sound intensity level for a 13.0 cm diameter speaker that has an efficiency of 2.06% is [tex]5.87*10^{-4}W[/tex]
