Question

Harry and Sally are sitting on opposite sides of a circus tent when an elephant trumpets a loud blast. If Harry experiences a sound intensity level of 67 dB and Sally experiences only 45 dB, what is the ratio of the distance between Sally and the elephant to the distance between Harry and the elephant?

Answer 1

To solve this problem we will start from the definition of the decibels which refers to the log relationship of the intensities, mathematically each decibel of the people can be described as follows

$dB_1 = 10log(\frac{I_1}{I_o})$

$dB_2 = 10log(\frac{I_2}{I_o})$

$dB_1 - dB_2 = 10log(\frac{I_1}{I_2})$

$\frac{67-45}{10} = log(\frac{I_1}{I_2})$

$log(\frac{I_1}{I_2}) = \frac{11}{5}$

For properties of the logarithm then we have that,

$\frac{I_1}{I_2} = 10^{11/5}$

$\frac{I_1}{I_2} = 158.48$

We have the relation

$\frac{r_2^2}{r_1^2} = 158.48$

$\frac{r_2}{r_1} = \sqrt{158.48}$

$\frac{r_2}{r_1} = 12.58$