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**Hint:**We will calculate the increase in intensity in power by using the sound intensity formula. Just remember, the sound intensity level is the level of the intensity of a sound relative to a reference value. Also, it is the logarithmic expression that is relative to a reference value of sound intensity.

**Complete step by step solution:**

We are given that

${P_1} = 20\,mW$(Power of the sound from the speaker)

${P_2} = 400\,mW$(Power of the sound increased)

As we all know that intensity is the power per unit area.

Now, For a given source, \[P\,\alpha \,I\].

Where $P$ is the power and $I$ is the intensity in the sound.

If ${L_1}$and ${L_2}$ are the initial and final level of loudness, then

${L_1} = 10\log (\dfrac{{{I_1}}}{{{I_0}}})$

And ${L_2} = 10\log (\dfrac{{{I_2}}}{{{I_0}}})$

$ \Rightarrow {L_2} - {L_1} = 10\log (\dfrac{{{I_2}}}{{{I_1}}})$

Therefore, an increase in loudness of sound can be calculated as

$\Delta L = 10\log (\dfrac{{{P_2}}}{{{P_1}}})$

\[ \Rightarrow \Delta L = 10\log (\dfrac{{400}}{{20}})\]

$ \Rightarrow \,\Delta L = 10\log (20)$

$\therefore \,$ $\Delta L = 13dB$

Therefore, the increase in power as compared to the original power is$13\,db$.

**Therefore, option (A) is the correct option.**

**Additional Information:**

As we know, the Sound intensity level is the level of intensity of sound which is relative to a reference value. It is denoted by ${L_I}$ and is expressed as

${L_I} = \dfrac{1}{2}\ln (\dfrac{I}{{{I_0}}})\,Np$

Or ${L_I} = {\log _{10}}(\dfrac{I}{{{I_0}}})\,B$

Or ${L_I} = 10\log (\dfrac{I}{{{I_0}}})\,dB$

Where ${L_I}$ is the sound intensity level, $I$ is the intensity of sound, and ${I_0}$ is the reference sound intensity.

Here,$Np$, $B$, and $dB$ are the units in which sound intensity level could be measured and these are abbreviated as $neper$, $bel$, and $decibel$.

**Note:**As we know, ${I_0}$ is the reference sound intensity and its value in the air is given by

${I_0} = 1\,pW/{m^2}$(Pico watts per meter square).

It is the least value of the sound intensity but it is hearable by the human ear at room conditions.

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