Question

How intense is 80 dB sound in comparison to 40 dB?
(A) ${10}^4$
(B) ${10}^2$
(C) $2$
(D) ${\displaystyle \frac{1}{2}}$

Answer 1

Hint: We are quite known with the term known as a decibel. It is a parameter to measure the intensity of sound or the signal by comparing it with a reference level on a logarithmic scale. In general, it is the measure of loudness. It measures the relative loudness of the sound in the form of ratios.

Complete step by step answer:
The formula for the loudness of sound is,
$$L=10{\log }_{10}\left({\displaystyle \frac{I}{I_0}}\right)$$
Here I0 is the intensity of sound for the reference level, and I is the intensity of sound at the given level. It is the ratio of relative measures of powers. The sound can also be approximated in terms of frequencies.
Let $L_1$ be the loudness for 40 dB sound and $L_2$ be the loudness for 80 dB sound. So, therefore, we can now represent these both as,
$L_1=10\log \left({\displaystyle \frac{I_1}{I_0}}\right)$
Here $I_1$ is the intensity of sound for 40 dB and $I_0$ is the intensity of sound for the reference level.
We will now substitute 40 dB $I_1$ to calculate the value of $L_1$, and hence we will get,
$$\begin{array}{l}40\mathrm{d}\mathrm{B}=10\log \left({\displaystyle \frac{I_1}{I_0}}\right)\\ {\displaystyle \frac{I_1}{I_0}}={10}^4.......(\mathrm{I})\end{array}$$
So we will now represent $L_2$ as,
$L_2=10\log \left({\displaystyle \frac{I_2}{I_0}}\right)$
Here $I_2$ is the intensity of sound for 40 dB and $I_0$ is the intensity of sound for the reference level.
We will now substitute 80 dB for $I_2$ to calculate the value of $L_2$ , and hence we will get,
$$\begin{array}{l}80\mathrm{d}\mathrm{B}=10\log \left({\displaystyle \frac{I_2}{I_0}}\right)\\ {\displaystyle \frac{I_2}{I_0}}={10}^8......(\mathrm{I}\mathrm{I})\end{array}$$
Now we will divide equation (I) and equation (II), and we will get,
$\begin{array}{l}{\displaystyle \frac{I_2}{I_1}}={\displaystyle \frac{{10}^8}{{10}^4}}\\ {\displaystyle \frac{I_2}{I_1}}={10}^4\end{array}$
Therefore, the 80 dB sound is ${10}^4$ times more intense than the 40 dB sound, and the correct option is (A).

Additional Information:
When talking about dogs, they can hear sound frequencies up to 50000Hz, and we have seen unique whistles that dogs can listen to but we cannot hear. We humans can listen to the sound intensity, not more than 120 decibels. I suppose the sound power is more, let’s say up to 170 dB, then the feeling of pain occurs in the human ear. Even the clear and comfortable perception of sound is up to 8000Hz. Beyond 8000 Hz, there is a pain and difficulty experienced in hearing. For old, The maximum limit of hearing for people under 50 years of age is 12000 Hz.

Note:Sound intensity can also be explained as the power of the sound per unit area. The decibel scale is the ratio of the intensity of actual sound intensity to that sound intensity, which has a threshold value.