91Ó°ÊÓ

The formula that relates the decibel level (dB) to the loudness (L) of a sound is given by: \(L = 10 \cdot log_{10}(I/I_0)\) where \(L\) is the sound level in decibels (dB), \(I\) is the intensity of the sound, and \(I_0\) is a reference intensity. The reference intensity (\(I_0\)) is usually taken to be the lowest intensity that the average human can hear, \(I_0 = 10^{-12} \ W/m^2\).

We are given the initial sound level of the television as 50 decibels. We can use this information and the formula from Step 1 to find the initial intensity of the TV: 50 = \(10 \cdot log_{10}(I/10^{-12})\) Divide both sides of the equation by 10: 5 = \(log_{10}(I/10^{-12})\) Exponentiate both sides of the equation with base 10 to get rid of the logarithm: \(10^5 = I/10^{-12}\) Multiply both sides by \(10^{-12}\) to solve for I: \(I = 10^5 \cdot 10^{-12} = 10^{-7} \ W/m^2\)

We are asked to find the decibel level that would make the television sound eight times as loud. To do this, first find the intensity of the television when it is eight times as loud: Since loudness is proportional to intensity, the new intensity is: \(I_{new} = 8 \cdot I = 8 \cdot 10^{-7} = 8 \cdot 10^{-7} \ W/m^2\)

Now we can use the formula from Step 1 and the new intensity from Step 3 to find the new decibel level: \(L_{new} = 10 \cdot log_{10}(I_{new}/I_0) = 10 \cdot log_{10}(8 \cdot 10^{-7}/10^{-12})\) To simplify the expression inside the logarithm, divide the numbers: \( L_{new} = 10 \cdot log_{10} (8 \cdot 10^5) \) Now, apply the logarithm properties: \( L_{new} = 10 \cdot (log_{10}(8) + log_{10}(10^5)) \) Since \(log_{10}(10^5) = 5\), we can rewrite the expression as: \( L_{new} = 10 \cdot (log_{10}(8) + 5) \) Now use the fact that \(log_{10}(8) \approx 0.9031\): \( L_{new} \approx 10 \cdot (0.9031 + 5) \) Multiply the resulting expression: \( L_{new} \approx 59.03 \) So, the new decibel level that would make the television sound eight times as loud is approximately 59.03 dB.