A hearing test is conducted on an aged person. It is found that her threshold of hearing is 20 decibels at $1 \mathrm{kHz}$ and it rises linearly with frequency to 60 decibels at $9 \mathrm{kHz}$. The minimum intensity of sound that the person can hear at $5 \mathrm{kHz}$ is-

• A

  10 times than that at $1 \mathrm{kHz}$.
• Correct answer
  B

  100 times than that at $1 \mathrm{kHz}$
• C

  $0.5$ times than that at $9 \mathrm{kHz}$
• D

  $0.05$ times than that at $9 \mathrm{kHz}$

Solution

at $5 \mathrm{KHz}$ Hearing capacity $=40 \mathrm{~dB}$ Intensity at $1 \mathrm{KHz}$ $$ \begin{array}{l} \beta=10 \log \left(\frac{\mathrm{I}}{\mathrm{I}_{0}}\right) \\ \mathrm{I}=\mathrm{I}_{0} 10^{\left(\frac{\beta}{10}\right)} \\ (\mathrm{I})_{1 \mathrm{KHz}}=\mathrm{I}_{0} 10^{(20 / 10)}=\mathrm{I}_{0}(10)^{2} \\ (\mathrm{I})_{5 \mathrm{KHz}}=\mathrm{I}_{0} 10^{(40 / 10)}=\mathrm{I}_{0}(10)^{4} \\ \frac{(\mathrm{I})_{1 \mathrm{KHz}}}{\mathrm{I}_{5 \mathrm{KHz}}}=\frac{1}{100} \end{array} $$

Hence, the correct answer is option: 'B'.