ANSWER
B. 139.6 dB
C. 100 W/m²
EXPLANATION
B. From the first part, we have that the intensity of the Seattle roar was 45.7088 W/m². Now, the fans in Kansas City want to create a roar twice as intense,
[tex]I=2\cdot45.7088W/m^2=91.4176W/m^2[/tex]
To find how many decibels this roar needs to be, we have to replace I with this value in the given equation,
[tex]L=10\log \mleft(\frac{91.4176}{10^{-12}}\mright)\approx10\cdot13.96=139.6dB[/tex]
The fans in Kansas City have to roar at 139.6 dB.
C. Now, their roar was 140 dB and we have to find the intensity. In the given equation, replace L with 140,
[tex]140=10\log \mleft(\frac{I}{I_o}\mright)[/tex]
And solve for I. Divide both sides by 10,
[tex]\frac{140}{10}=\log \mleft(\frac{I}{I_o}\mright)[/tex]
Rise 10 to the exponent of each side. By the rule of the exponent of the base of a logarithm, the result on the right side of the equation is what is in the parenthesis,
[tex]\begin{gathered} 10^{\frac{140}{10}}=10^{\log (\frac{I}{I_o})} \\ \\ 10^{\frac{140}{10}}=\frac{I}{I_o} \end{gathered}[/tex]
Finally, multiply both sides by I₀,
[tex]I=I_o\cdot10^{\frac{140}{10}}[/tex]
Replace the value of Io given and solve,
[tex]I=10^{-12}\cdot10^{\frac{140}{10}}=100W/m^2[/tex]
Hence, the intensity of the roar of the fans in Kansas City was 100 W/m².
