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Chapter 16: Problem 2

A sound level of 50 decibels is a) 2.5 times as intense as a sound of 20 decibels. b) 6.25 times as intense as a sound of 20 decibels. c) 10 times as intense as a sound of 20 decibels. d) 100 times as intense as a sound of 20 decibels. e) 1000 times as intense as a sound of 20 decibels.

Short Answer

Expert verified
Answer: The intensity of a 50-decibel sound is 1000 times as intense as a sound of 20 decibels.

Step by step solution

01

Write the given values and formula

We know that: - A 50-decibel sound has a decibel level of 50 dB - A 20-decibel sound has a decibel level of 20 dB The formula to convert decibel levels to intensities is: \( dB = 10 \times log_{10}(\frac{I}{I_0}) \)
02

Find the ratio of intensities#

First, let's consider the 50-decibel sound. Let \(I_1\) be its intensity. Using the formula, we have: \( 50 = 10 \times log_{10}(\frac{I_1}{I_0}) \) Now consider the 20-decibel sound. Let \(I_2\) be its intensity. Using the formula, we have: \( 20 = 10 \times log_{10}(\frac{I_2}{I_0}) \) We need to find the ratio \(\frac{I_1}{I_2}\).
03

Solve the equations for \(I_1\) and \(I_2\)

First, divide both sides of the equations by 10: \( 5 = log_{10}(\frac{I_1}{I_0}) \) \( 2 = log_{10}(\frac{I_2}{I_0}) \) Now, eliminate the logarithms using the property \(a = log_{10}(b) \Rightarrow b = 10^a\): \( I_1 = I_0 \times 10^5 \) \( I_2 = I_0 \times 10^2 \)
04

Calculate the intensity ratio

The intensity ratio between the two sounds can be found by dividing \(I_1\) by \(I_2\): \( \frac{I_1}{I_2} = \frac{I_0 \times 10^5}{I_0 \times 10^2} \) Simplify the expression by canceling the \(I_0\) terms: \( \frac{I_1}{I_2} = \frac{10^5}{10^2} \) Now, use the property \( \frac{a^b}{a^c} = a^{b-c}\) for exponents: \( \frac{I_1}{I_2} = 10^{(5-2)} \) \( \frac{I_1}{I_2} = 10^3 \) Thus, the intensity of a 50-decibel sound is 1000 times as intense as a sound of 20 decibels.
05

Select the correct answer

The intensity ratio \(\frac{I_1}{I_2}\) is 1000, which corresponds to option (e). Hence, the correct answer is: e) 1000 times as intense as a sound of 20 decibels.

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Most popular questions from this chapter

A bugle can be represented by a cylindrical pipe of length \(L=1.35 \mathrm{~m} .\) Since the ends are open, the standing waves produced in the bugle have antinodes at the open ends, where the air molecules move back and forth the most. Calculate the longest three wavelengths of standing waves inside the bugle. Also calculate the three lowest frequencies and the three longest wavelengths of the sound that is produced in the air around the bugle.

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