A company with loud machinery needs to cut its sound intensity to 44% of its original level. By how many decibels would the loudness be reduced? Use the formula L=101logI/Io. Round to the nearest hundredth. (1 point)3.57 decibels
The Correct Answer and Explanation is:
To determine the reduction in decibels when a company’s sound intensity is cut to 44% of its original level, we use the decibel formula: L=10log(II0)L = 10 \log \left(\frac{I}{I_0}\right)
Where:
- LL is the sound level in decibels (dB),
- II is the final sound intensity,
- I0I_0 is the original sound intensity,
- log\log is the base-10 logarithm.
Step 1: Understand the problem
The problem says the new sound intensity is 44% of the original, so: II0=0.44\frac{I}{I_0} = 0.44
We want to know how many decibels the loudness is reduced by — in other words, how much less intense it is in decibels.
Step 2: Use the formula to calculate decibel reduction
We plug the ratio II0=0.44\frac{I}{I_0} = 0.44 into the formula: L=10log(0.44)L = 10 \log(0.44)
Now compute the logarithm: log(0.44)≈−0.3565\log(0.44) \approx -0.3565
Multiply by 10: L=10×(−0.3565)=−3.565L = 10 \times (-0.3565) = -3.565
Rounding to the nearest hundredth: L≈−3.57 decibelsL \approx -3.57 \text{ decibels}
The negative sign indicates a reduction in loudness.
Step 3: Interpret the result
The sound has decreased by approximately 3.57 decibels. This means that if the original sound level was at a certain dB, reducing its intensity to 44% of the original will reduce the perceived loudness by 3.57 dB.
Remember, decibels are a logarithmic scale, so even a few decibels’ change represents a significant difference in intensity. A decrease of 3.57 dB is quite noticeable in an environment, especially in places where noise control is important for health and safety.