Question

The % increase in the amplitude of sound is 20%, then % change in loudness is _________.

• A. 11
• B. 22
• C. 33
• D. 44

Answer 1

The correct option is D

44

Step 1: Given Data

Let A₁ be the initial amplitude of the sound.

$A_1=A$

Now consider loudness as L, So L₁=L

Let A₂ be the amplitude of the sound after the 20% increase in the amplitude of the sound.

$A_2=20\%increaseinA=100+20\%=\frac{120}{100}A$

Now consider loudness as L₂=?

Step 2: Calculations

From the relation between loudness and amplitude we have,

$Loudness\propto {\left(Amplitude\right)}^2$

So,

$$\begin{gathered} L_1\alpha {A_1}^2 \\ and \\ {L}_2\alpha {A_2}^2 \end{gathered}$$

Taking the ratios

$$\begin{gathered} \frac{L_2}{L_1}={\left(\frac{A_2}{A_1}\right)}^2{L}_2={\left(\frac{A_2}{A_1}\right)}^2{L}_1 \\ {L}_2={\left(\frac{120A}{100A}\right)}^{2}L \\ {L}_2=1.44L \end{gathered}$$

Change in loudness

$$\begin{gathered} ∆L=L_2-L_1 \\ =1.44L-L \\ ∆L=0.44L=\frac{44}{100}=44\%L \end{gathered}$$

% change in loudness is 44%

Hence, the correct option is (D)