Question

Using your knowledge of exponential and logarithmic functions and properties, what is the intensity of a fire alarm that has a sound level of 120 decibels?
A.
1.0x10^-12 watts/m^2
B.
1.0x10^0 watts/m^2
C.
12 watts/m^2
D.
1.10x10^2 watts/m^2

Answer 1

Option B:

$I=1.0\times\ 10^{0} \ \text {watts}/ \text m^2}$

Solution:

Given sound level = 120 decibel

To find the intensity of a fire alarm:

$\beta=10\log\left(\frac{I}{I_0} \right)$

where $I_0=1\times10^{-12}\ \text {watts}/ \text m^2}$

Step 1: First divide the decibel level by 10.

120 ÷ 10 = 12

Step 2: Use that value in the exponent of the ratio with base 10.

$10^{12}$

Step 3: Use that power of twelve to find the intensity in Watts per square meter.

$10^{12}=\left(\frac{I}{I_0} \right)$

$10^{12}=\left(\frac{I}{1\times10^{-12}\ \text {watts}/ \text m^2} \right)$

Now, do the cross multiplication,

$I=10^{12}\times1\times\ 10^{-12} \ \text {watts}/ \text m^2}$

$I=1\times\ 10^{12-12} \ \text {watts}/ \text m^2}$

$I=1\times\ 10^{0} \ \text {watts}/ \text m^2}$

$I=1.0\times\ 10^{0} \ \text {watts}/ \text m^2}$

Option B is the correct answer.

Hence $I=1.0\times\ 10^{0} \ \text {watts}/ \text m^2}$.