Problem 117 - College Algebra

Question

The loudness level of a sound, \(D,\) in decibels, is given by the formula $$D=10 \log \left(10^{12} I\right)$$ where \(I\) is the intensity of the sound, in watts per meter \(^{2}\). Decibel levels range from \(0,\) a barely audible sound, to \(160,\) a sound resulting in a ruptured eardrum. (Any exposure to sounds of 130 decibels or higher puts a person at immediate risk for hearing damage.) Use the formula to solve. The sound of a blue whale can be heard 500 miles away, reaching an intensity of \(6.3 \times 10^{6}\) watts per meter\(^{2}\). Determine the decibel level of this sound. At close range, can the sound of a blue whale rupture the human eardrum?

Step-by-Step Solution

Answer

The decibel level of the sound produced by a blue whale is 188 decibels. Therefore, at close range, the sound of a blue whale can rupture the human eardrum, as it is higher than the 160 decibel rupture threshold.

Step 1: Insert information into formula

The first step is to substitute the given intensity of the whale's sound, \(6.3 x 10^{6}\), into the formula for D, the decibel level. The resulting equation is \(D = 10 \log \left(10^{12} x 6.3 \times 10^{6}\right)\)

Step 2: Solve for D

Next, simplify the equation inside the logarithm and solve for D. After simplifying, the new formula becomes: \(D = 10 \log \left(6.3 \times 10^{18} \right)\). Apply the logarithmic rule \(\log(ab) = \log(a) + \log(b)\) to split the logarithm, which leads to: \(D = 10 (\log 6.3 + \log 10^{18})). Conveniently, \(\log 10^{18} = 18\). Therefore, \(D = 10 (\log 6.3 + 18)\) is our new equation.

Step 3: Calculate the value of D

The final step to find the decibel level D is to calculate the value of the logarithm and multiply by 10. Using a calculator, \(\log 6.3 \approx 0.8\). Substituting this into the equation gives \(D = 10 (0.8 + 18)= 10(18.8) = 188\).

Key Concepts

Decibel Levels

Decibels, abbreviated as dB, are units that measure the intensity or loudness of sound. This unit helps us quantify the magnitude of sounds in a manner that is relatable, since human ears do not perceive sound linearly. The decibel system is logarithmic, which means each 10 dB increase represents a tenfold increase in sound intensity. This is different from linear scales where each unit increase is additive.

• A sound measuring 0 dB is just at the threshold of human hearing.
• Normal conversation typically falls around 60 dB.
• Sounds above 120 dB may cause discomfort or even pain.
• 130 dB and higher can cause immediate hearing damage.

Understanding the decibel system is crucial for determining when sound levels become potentially harmful. In the exercise with the blue whale, when calculated, the sound reaches 188 dB. This calculation indicates an extremely intense sound level, far above the threshold where damage might occur.

Sound Intensity

Sound intensity is a measure of the power of sound waves per unit area, often described in watts per meter squared (W/m²). It gives us an idea about how much sound energy passes through a specific area.
In the given exercise, the intensity of a blue whale's call is incredibly high at \(6.3 \times 10^{6}\) W/m². This intensity makes it one of the loudest noises produced by any animal.

• Sound intensity is directly related to decibels by the formula provided in the exercise: \[ D = 10 \log \left(10^{12} I\right) \]
• The logarithmic relationship shows how increases in intensity lead to a significant increase in the perceived loudness.

This relationship helps us understand how seemingly small increases in intensity can lead to large increases in decibel levels, emphasizing the non-linear nature of how we perceive sound.

Logarithmic Rules

Logarithmic functions are powerful mathematical tools used to simplify complex calculations. Logarithms express the power to which a number, called the base, must be raised to produce a given number. In most sound-related problems, the common logarithm (base 10) is used.
Some key logarithmic properties were used in solving the exercise:

• The property \( \log(ab) = \log(a) + \log(b) \) allows us to break down complex products into sums, simplifying calculations.
• For powers of ten, such as \( \log(10^{18}) = 18 \), logarithms make it easier to manage large numbers.

Using these rules, the exercise simplifies the whale's sound intensity-derived formula and effectively calculates the decibel level. Being comfortable with these logarithmic properties is vital in handling real-world applications involving exponential growth or decay, such as sound intensity and decibel calculations.