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podryga [215]
3 years ago
10

The sound intensity of a typical vacuum cleaner is 80 dB (decibels). Sound intensity of 160 dB will burst an eardrum. How many t

imes louder is the sound that will burst an eardrum than the sound of a vacuum cleaner?
Mathematics
2 answers:
NemiM [27]3 years ago
5 0

Answer:

Ten thousand times as loud.

Nataly [62]3 years ago
4 0

Answer:

Using logarithmic rule:

\log \frac{m}{n} = \log m -\log n

\log_b b^a = a

\log_b x = a then x = b^a

Given that,

Sound intensity of a typical vacuum cleaner = 80 dB

Sound intensity of burst of eardrum = 160 dB

Let, I₁ and I₂ b e the intensity of typical vacuum and intensity of eardrum.

Use the formula:

1 dB = 10\log_{10} (\frac{I}{I_0})

As per the statement

The sound intensity of a typical vacuum cleaner is 80 dB (decibels).

⇒ 80 dB= 10\log_{10} (\frac{I_1}{10^{-12}})

Divide both sides by 10 we have;

8 dB= \log_{10} (\frac{I_1}{10^{-12}})

Apply the logarithmic rules:

then;

8 = \log_{10} I_1 - \log_{10} 10^{-12}

⇒8 = \log_{10} I_1 + 12

Subtract 12 from both sides we have;

-4= \log_{10} I_1

⇒10^{-4} = I_1                                    ......[A}

It is also given that: Sound intensity of 160 dB will burst an eardrum.

⇒ 160 dB= 10\log_{10} (\frac{I_2}{10^{-12}})  

Divide both sides by 10 we have;

16 dB= \log_{10} (\frac{I_2}{10^{-12}})

Apply the logarithmic rules:

16 = \log_{10} I_2 - \log_{10} 10^{-12}

⇒16 = \log_{10} I_2 + 12

Subtract 12 from both sides we have;

⇒4 = \log_{10} I_2

⇒10^{4} = I_2                                              ......[B]

We have to find how many times louder is the sound that will burst an eardrum than the sound of a vacuum cleaner.

\dfrac{I_2}{I_1} = \dfrac{10^4}{10^{-4}} = 10^8

⇒I_2 = 10^8 \times I_1

Therefore, 10^8 times louder is the sound that will burst an eardrum than the sound of a vacuum cleaner

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