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Chapter 3: Problem 48

Suppose a radio is playing loudly at a sound level of 80 decibels. What decibel level would make the radio sound one-fourth as loud?

Short Answer

Expert verified
The new decibel level that would make the radio sound one-fourth as loud is approximately 73.98 decibels.

Step by step solution

01

Calculate the intensity of the original sound

We are given the original decibel level, which is 80 dB. Using the formula d = 10 * log10(I/I0), we can solve for I: 80 = 10 * log10(I/I0) -> I = I0 * 10^(80/10)
02

Find the intensity of the sound that is one-fourth as loud

We want to find the intensity of the sound that is one-fourth as loud as the original. The intensity ratio for the perceived loudness is described as follows: (I_new/I0) / (I/I0) = 1/4 We can now substitute the value of I found in Step 1: (I_new/I0) / (10^(80/10)) = 1/4 Solving for I_new: I_new = I0 * 10^(80/10) * (1/4)
03

Calculate the new decibel level

Now that we have the new intensity I_new, we can use the decibel formula to find the new decibel level d_new: d_new = 10 * log10(I_new/I0) Plug in the value for I_new from Step 2: d_new = 10 * log10((I0 * 10^(80/10) * (1/4))/I0) We can cancel I0 from the numerator and denominator: d_new = 10 * log10(10^(80/10) * (1/4)) Now simplify the expression inside the logarithm: d_new = 10 * log10(10^8 * 1/4) d_new = 10 * log10(10^8 / 4) d_new = 10 * log10(10^8) - 10 * log10(4) Using the properties of logarithms, we have: d_new = 8 * 10 - 10 * log10(2^2) d_new = 80 - 20 * log10(2)
04

Approximate the new decibel level

Now, we can approximate the value of log10(2): log10(2) 鈮� 0.301 Substitute this value back into the equation: d_new = 80 - 20 * 0.301 d_new 鈮� 80 - 6.02 d_new 鈮� 73.98 So, the new decibel level that would make the radio sound one-fourth as loud is approximately 73.98 decibels.

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