Problem 105 - Pearson Physics

Question

A machine shop has 120 equally noisy machines that together produce an intensity level of \(92 \mathrm{~dB}\). If the intensity level must be reduced to \(82 \mathrm{~dB}\), how many machines must be turned off?

Step-by-Step Solution

Answer

Turn off 108 machines to reduce the intensity level to 82 dB.

Step 1: Understanding Decibels, dB

Decibels (dB) measure the intensity level of sounds. The formula to convert intensity (\(I\)) of sound to decibels is given by\[L = 10 \times \log_{10}\left(\frac{I}{I_0}\right)\]where \(L\) is the intensity level in dB, \(I\) is the intensity of the sound, and \(I_0\) is the reference intensity, typically \(10^{-12} \ ext{watts/m}^2\). In this problem, the quantity of machines influences the measured intensity level.

Step 2: Calculate Initial Intensity Level

Given that 120 machines produce an intensity level of 92 dB, the sum of these machines results in a large intensity \(I_{initial}\). The formula\[92 = 10 \times \log_{10}\left(\frac{I_{initial}}{I_0}\right)\]can be rearranged to find \(I_{initial}\).

Step 3: Calculate Required Intensity Level

We want to decrease the intensity level to 82 dB. If we have \(n\) machines running, the new intensity becomes \(I_{new}\), and the equation becomes\[82 = 10 \times \log_{10}\left(\frac{I_{new}}{I_0}\right)\].

Step 4: Relate Intensities to Number of Machines

The intensity level relates directly to the number of machines running. Given that the intensity level is proportional to the intensity produced per machine, we set\[ \log_{10}(I_{new}) = \log_{10}(I_{initial}) - \log_{10}(\text{Reduction factor})\]. The reduction in the number of machines will lower \(I_{initial}\) to \(I_{new}\).

Step 5: Calculate Reduction Factor

Using the decibel formula difference:\[(92 - 82) = 10 \times \log_{10}(\text{Reduction factor})\]Solve this to find \[\text{Reduction factor} = 10^{(10/10)} = 10\]. This means that the intensity should be 1/10 of the original.

Step 6: Determine Number of Machines to Turn Off

If 120 machines produce \(I_{initial}\), to reduce the intensity to 1/10 of \(I_{initial}\), only \((1/10)\) of the machines should be running\[ = \frac{120}{10} = 12\]. Thus, only 12 machines need to remain on, meaning \(120 - 12 = 108\) machines should be turned off.

Key Concepts

Decibels (dB)

Decibels, abbreviated as dB, are units used to measure the intensity of sound. The decibel scale is logarithmic, meaning each step on the scale represents a tenfold change in intensity.
This allows us to cover a vast range of sound levels with manageable numbers. In simpler terms, a small change in the dB value reflects a large change in intensity.
To calculate the intensity in decibels, we use the formula: \[ L = 10 \times \log_{10}\left(\frac{I}{I_0}\right) \] Here, \(L\) is the sound level in dB, \(I\) is the intensity of the sound, and \(I_0\) is a reference intensity, typically \(10^{-12} \text{watts/m}^2\) which represents the faintest sound the human ear can detect.
This concept is crucial when working with many sources of sound, such as machines, since even a noise from multiple identical machines can add up to significant levels.

Sound Intensity Calculation

Calculating sound intensity involves determining how the number of sources (like machines) contributes to the total sound measured.
When many machines operate concurrently, their combined intensity can be quite high.
In our case, 120 machines together produce 92 dB.
To find the initial sound intensity, we rearrange the formula: \[ 92 = 10 \times \log_{10}\left(\frac{I_{initial}}{I_0}\right) \] Solving this gives us \(I_{initial}\), which represents the power per unit area of the sound from all the machines.
When calculating reductions, understanding how changes in intensity affect the dB level is essential. Our goal here is to reduce the dB level from 92 to 82. This involves understanding the relationship between dB change and intensity, and how turning some machines off achieves the desired reduction.

Machine Noise Reduction

Reducing machine noise effectively involves a clear understanding of how the number of machines affects the surrounding sound environment. In this scenario, noise reduction is achieved by turning off some machines.
The key to reducing the noise level from 92 dB to 82 dB lies in understanding the logarithmic nature of the dB scale. We need to reduce the intensity to one-tenth of its original level.
This is calculated using the formula difference:\[(92 - 82) = 10 \times \log_{10}(\text{Reduction factor})\]Computing this yields a reduction factor of 10.Therefore, if all 120 machines produce the initial intensity, to achieve a 1/10 intensity level, we only need 12 machines running. Hence, by turning off 108 machines, we can significantly reduce the noise level, making it more manageable and safer for the environment.