Problem 35 - Contemporary Precalculus

Question

Deal with the energy intensity i of a sound, which is related to the loudness of the sound by the function \(L(i)=10 \cdot \log \left(i / i_{0}\right),\) where \(i_{0}\) is the minimum intensity detectable by the human ear and \(L(i)\) is measured in decibels. Find the decibel measure of the sound. Victoria Falls in Africa (intensity is 10 billion times \(i_{0}\) ).

Step-by-Step Solution

Answer

Answer: The loudness of the sound at Victoria Falls is 100 dB.

Step 1: Understand the given function for loudness

The loudness L(i) is given by the function \(L(i)=10 \cdot \log \left(\frac{i}{i_{0}}\right)\). In this formula, i is the energy intensity of the sound, and \(i_0\) is the minimum intensity detectable by the human ear. The loudness is measured in decibels.

Step 2: Substitute the intensity at Victoria Falls

We know that Victoria Falls has an intensity 10 billion times \(i_{0}\). So the intensity i at Victoria Falls is \(i = 10\cdot 10^9 \cdot i_0\). Substitute this value of i in the loudness function. $$L(i) = 10 \cdot \log \left(\frac{10\cdot 10^9 \cdot i_0}{i_0}\right)$$

Step 3: Simplify the equation

In the equation, we can see that \(i_0\) in both numerator and denominator cancels out. Simplify the equation. $$L(i) = 10 \cdot \log (10 \cdot 10^9)$$

Step 4: Calculate the loudness in decibels

Now we need to calculate the decimal logarithm of \((10 \cdot 10^9)\) and multiply the result by 10. $$L(i) = 10 \cdot \log (10 \cdot 10^9) = 10 \cdot (10) = 100$$ The loudness of the sound at Victoria Falls has a decibel measure of 100 dB.

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions and play a critical role in various scientific disciplines, including acoustics. In simple terms, a logarithm tells us what exponent we need to raise a base number to get another number. For instance, if we have a base of 10 and want to know the logarithm of 100, we calculate \( \log_{10}(100) \), which equals 2, because \(10^2 = 100\).

It's important to grasp that log scales are not linear, but rather they compress a wide range of values into a smaller scale. This is especially useful when dealing with measures that have a big variance, like sound intensity. In our exercise, the logarithmic function helps to convert the vast range of sound intensities into a more manageable scale of decibels. Logarithms make it possible to compare and understand very big or small numbers, like the sound intensity of Victoria Falls, which is significantly higher than the quietest sound a human can hear.

Decibels

Decibels (dB) are the units used to measure the intensity of a sound, and they are based on a logarithmic scale. This measurement reflects the level of sound pressure relative to a reference value. The reference point, denoted by \(i_0\), in our context is the lowest sound intensity that the average human ear can detect, and it's standardized at \(10^{-12} \) watts per square meter.

Because the decibel scale is logarithmic, every increase of 10 dB represents a tenfold increase in intensity. Thus, a sound measuring 20 dB is 10 times more intense than one at 10 dB, while a sound at 30 dB is 100 times more intense than one at 10 dB. Such a scale allows us to manage the vast range of sound intensities. This concept is why we can describe the enormous sound intensity at Victoria Falls using a manageable figure, like 100 dB, rather than an unfathomably large number.

Loudness Level Calculation

The calculation of loudness levels in decibels involves using the function \( L(i) = 10 \cdot \log \left(\frac{i}{i_0}\right) \). In this formula, \( i \) represents the intensity of the sound in question, and \( i_0 \) is the threshold of hearing, or the sound intensity of the softest sound that a typical human ear can detect. Decibels quantify the relative loudness of sounds, with higher dB values representing louder sounds.

To calculate the loudness level of the sound at Victoria Falls, we substitute its intensity into the formula, which is 10 billion times greater than \(i_0\). Using properties of logarithms, we simplify the formula. This leads us to a calculation of 100 dB, meaning the sound of Victoria Falls is significantly loud, especially when compared to the threshold of human hearing. Understanding this calculation is essential for accurately assessing the intensity of different sounds and their impact on our hearing.