Question

Decibel $\left( {db} \right)$ is a unit of loudness of sound. It is defined in a manner such that when amplitude of sound is multiplied by a factor of $\sqrt {10}$, the decibel level increases by $10$ units. Loud music of $70dB$ is being played at a function. To reduce the loudness to a level of $30dB$ , the amplitude of the instrument playing music to be reduced by a factor of
A. $0$
B. $10\sqrt {10}$
C. $100$
D. $100\sqrt {100}$

Answer 1

In this question, we can solve the equation $dB = 10\log {a_0}^2 - 10\log {a_1}^2$. After then we can assume that the amplitude is changed by the factor $n$. Now, we can solve the above equation for $n$.

Complete step by step solution: -
We know that the loudness of sound is given by equation
$dB = 10\log \left( {\dfrac{{{I_0}}}{{{I_1}}}} \right)$
$\Rightarrow dB = 10\log {I_0} - 10\log {I_1}$

We know that intensity is directly proportional to the square of the amplitude i.e.
$I \propto {a^2}$
So,
$dB = 10\log {a_0}^2 - 10\log {a_1}^2$

According to the question, if amplitude of sound is multiplied by a factor of$\sqrt {10}$, the decibel level increases by $10$ units. So,
$d{B^1} = 10\log {\left( {\sqrt {10} } \right)^2} - 10\log {a_1}^2$
$\Rightarrow d{B^1} = 10\log 10 - 10\log {a_1}^2$
$\Rightarrow d{B^1} = 10 - 10\log {a_1}^2$
$\Rightarrow d{B^1} = 10 - dB$
Where $$dB = 10\log {a_1}^2$

Now, if [70dB$is being played at a function and if it is reduced to a level of$30dB$$ , then let the amplitude of the instrument playing music be reduced by a factor of $n$.
So,
$d{B^1} = 10\log \left( {\dfrac{{{a_1}^2}}{{{n^2}}}} \right)$
$\Rightarrow d{B^1} = 10\log {a_1}^2 - 10\log {n^2}$
$\Rightarrow d{B^1} = dB - 20\log n$
$\Rightarrow d{B^1} - dB = 20\log n$
$\Rightarrow 70 - 30 = 20\log n$
$\Rightarrow 40 = 20\log n$
$\Rightarrow 2 = \log n$
$\Rightarrow n = {10^2}$
$\Rightarrow n = 100$

So, loud music of $70dB$ is being played at a function. To reduce the loudness to a level of$30dB$ , the amplitude of the instrument playing music to be reduced by a factor of $100$ .

Hence, option C is correct.

Additional information: -
Decibel is a logarithmic unit which is used to measure the loudness. It is used in electronics, signals and communications. Decibel is a logarithmic way of describing ratios of power, sound pressure, voltage, intensity, etc. Generally, it is used to measure the loudness of the sound. The level $0dB$ occurs when the intensity of the sound is equal to the reference level of the sound.

Note:
In this question, we have kept in mind that $d{B^1}$ is the difference in decibel. We have to remember the calculations of logarithmic also such as $\log 10 = 1$ and $\log 1 = 0$.