The combined sound pressure level measured at a point in a production bench due to one dumper and one shovel is 95 dB(A). If the sound pressure level of shovel alone is 90 dB(A), t... The combined sound pressure level measured at a point in a production bench due to one dumper and one shovel is 95 dB(A). If the sound pressure level of shovel alone is 90 dB(A), the sound pressure level of the dumper alone, in dB(A), at the same point is __________ (round off up to 2 decimals).
Understand the Problem
The question is asking to calculate the sound pressure level of a dumper given the combined sound pressure level of a dumper and a shovel, as well as the sound pressure level of the shovel alone. This requires understanding of how sound levels in decibels combine.
Answer
$93.36 \, \text{dB(A)}$
Answer for screen readers
The sound pressure level of the dumper alone, in dB(A), at the same point is $93.36 , \text{dB(A)}$.
Steps to Solve
- Understanding the problem setup
We have three parameters:
- The combined sound pressure level of the dumper and shovel, ( L_{total} = 95 , \text{dB(A)} )
- The sound pressure level of the shovel alone, ( L_{shovel} = 90 , \text{dB(A)} )
- We need to find the sound pressure level of the dumper, denoted as ( L_{dumper} ).
- Using the formula for combining sound levels
The formula for combining sound levels in decibels is given by: $$ L_{total} = 10 \log_{10}(10^{\frac{L_{dumper}}{10}} + 10^{\frac{L_{shovel}}{10}}) $$
- Rearranging the equation to solve for ( L_{dumper} )
Substituting ( L_{total} ) and ( L_{shovel} ): $$ 95 = 10 \log_{10}(10^{\frac{L_{dumper}}{10}} + 10^{\frac{90}{10}}) $$
- Simplifying the equation
Calculate ( 10^{\frac{90}{10}} ): $$ 10^{\frac{90}{10}} = 10^9 = 1,000,000,000 $$
Thus, the equation becomes: $$ 95 = 10 \log_{10}(10^{\frac{L_{dumper}}{10}} + 1,000,000,000) $$
- Solving for ( L_{dumper} )
Divide by 10: $$ 9.5 = \log_{10}(10^{\frac{L_{dumper}}{10}} + 1,000,000,000) $$
Convert from logarithmic form: $$ 10^{9.5} = 10^{\frac{L_{dumper}}{10}} + 1,000,000,000 $$
Calculate ( 10^{9.5} ) (approximately ( 3,162,277,660.17 )): $$ 3,162,277,660.17 = 10^{\frac{L_{dumper}}{10}} + 1,000,000,000 $$
- Isolating ( 10^{\frac{L_{dumper}}{10}} )
Subtract ( 1,000,000,000 ): $$ 3,162,277,660.17 - 1,000,000,000 = 10^{\frac{L_{dumper}}{10}} $$ $$ 2,162,277,660.17 = 10^{\frac{L_{dumper}}{10}} $$
- Taking the logarithm again to find ( L_{dumper} )
Apply logarithm: $$ \frac{L_{dumper}}{10} = \log_{10}(2,162,277,660.17) $$
- Calculating ( L_{dumper} )
Calculate: $$ \log_{10}(2,162,277,660.17) \approx 9.336 $$ Now multiply by 10: $$ L_{dumper} = 9.336 \times 10 = 93.36 $$
The sound pressure level of the dumper alone, in dB(A), at the same point is $93.36 , \text{dB(A)}$.
More Information
The calculated sound pressure level indicates how loud the dumper is compared to the shovel and the overall sound environment. Understanding sound levels in decibels is crucial in contexts like mining where machinery noise needs to be managed for safety and compliance.
Tips
- Ignoring logarithmic properties: When combining decibels, remember to convert to linear scale first.
- Not properly isolating variables: It’s important to rearrange equations carefully and track your steps linearly.
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