Students were tasked with the following problem:
The sound intensity level (loudness) $y$ is measured on a logarithmic scale using the unit decibel ($\mathrm{dB}$), and is expressed by the formula: $$ y=10\log\frac{x}{x_0} $$ where $x$ is the sound intensity (measured in $W\cdot m^{-2}$) and $x_0$ is the sound intensity of the faintest sound a human ear can detect ($x_0=10^{-12}\, W\cdot m^{-2}$).
The noise level of a washing machine is about $50$ decibels during washing, but it can increase to $75$ decibels during spinning.
How many times more intense is the sound of the washing machine during spinning compared to the sound intensity during washing?
Alice: This is a very easy question. The sound during spinning is $1.5$ times more intense because: $$ \frac{75}{50}=\frac32=1.5 $$
Bob: To solve the task, we must substitute the values into the given formula: $$ \frac{y_2}{y_1}=\frac{10\log\frac{75}{x_0}}{10\log\frac{50}{x_0}} $$ Then, we cancel out the $10$ in the numerator and denominator and use the rule that the ratio of logarithms equals the difference of logarithms. For $x_0$, we substitute $1$: $$ \frac{y_2}{y_1}=\frac{10\log\frac{75}{x_0}}{10\log\frac{50}{x_0}}=\log 75- \log 50=0.176 $$
So the sound during spinning is about $18\%$ more intense than the sound during washing.
Cecilia: To solve the task, we substitute into the given formula: $$ \frac{y_2}{y_1}=\frac{10\log\frac{75}{x_0}}{10\log\frac{50}{x_0}} $$ We cancel the $10$ and substitute $x_0=1$: $$ \frac{y_2}{y_1}=\frac{10\log\frac{75}{x_0}}{10\log\frac{50}{x_0}}=\frac{\log75}{\log50}=1.104 $$ So the sound during spinning is approximately $110\%$ more intense than during washing. This means the sound is more than twice as strong.
David: We need to substitute into the given formula, where $y$ represents the sound level in decibels: $$ \begin{aligned} 50=10\log\frac{x_1}{x_0} \cr 5=\log\frac{x_1}{x_0} \cr 10^5=\frac{x_1}{x_0} \cr x_1=10^5\cdot x_0 \end{aligned} $$ And similarly: $$ x_2=10^{7.5}\cdot x_0 $$ Therefore: $$ \frac{x_2}{x_1}=\frac{10^{7.5}\cdot x_0}{10^{5}\cdot x_0}=\frac{10^{7.5}}{10^5}=10^{2.5} \doteq 316 $$
So, the sound during spinning is approximately $316$ times more intense than during washing.
Which student solved the problem correctly?
Alice
Bob
Cecilia
David
No one
Here is a brief explanation of a difference between sound intensity and loudness:
Sound intensity is a physical quantity that describes how much energy a sound wave transfers through a unit area each second. It is measured in watts per square meter ($\mathrm{W}/{m}^2$) and depends only on the source and distance-not on how we perceive the sound.
To relate intensity to how we hear sound, we use the sound intensity level (loudness), measured in decibels ($\mathrm{dB}$). This logarithmic scale compresses a wide range of intensities into manageable numbers and better reflects how we perceive differences in loudness.