document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192174>Question 1192174</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The intensity level of sound is measured on a logarithmic scale. The intensity level, β, of sound is defined in the following equation β=10log I/Io where β is measured in decibels, and Io is the intensity of a reference level. The reference level usually taken is the \"threshold of hearing\", which is 1.0x10^-12 W/m^2.\r<BR>\n" );
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document.write( "a) What is the intensity level of the threshold of hearing?<BR>\n" );
document.write( "b) What is the intensity level of a whisper if the intensity is 1.0x10^-10 W/m^2?<BR>\n" );
document.write( "c) How much louder does a siren at 30 m away, with an intensity of 1.0x10^-2 W/m^2, sound compared to a whisper? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849074 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849074\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **Understanding Sound Intensity and Decibels**\r<BR>\n" );
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document.write( "Sound intensity is measured on a logarithmic scale using decibels (dB). This scale is based on the equation:\r<BR>\n" );
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document.write( "β = 10 log (I / I₀)\r<BR>\n" );
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document.write( "where:\r<BR>\n" );
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document.write( "* β is the sound intensity level in decibels (dB)<BR>\n" );
document.write( "* I is the sound intensity in watts per square meter (W/m²)<BR>\n" );
document.write( "* I₀ is the reference intensity, usually the threshold of hearing (1.0 x 10⁻¹² W/m²)\r<BR>\n" );
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document.write( "**Calculations**\r<BR>\n" );
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document.write( "**a) Threshold of Hearing**\r<BR>\n" );
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document.write( "The threshold of hearing is our reference point (I₀ = 1.0 x 10⁻¹² W/m²). Plugging this into the equation:\r<BR>\n" );
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document.write( "β = 10 log (1.0 x 10⁻¹² / 1.0 x 10⁻¹²) = 10 log (1) = 0 dB\r<BR>\n" );
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document.write( "**The intensity level of the threshold of hearing is 0 dB.**\r<BR>\n" );
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document.write( "**b) Whisper**\r<BR>\n" );
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document.write( "A whisper has an intensity of 1.0 x 10⁻¹⁰ W/m².  Calculating the decibel level:\r<BR>\n" );
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document.write( "β = 10 log (1.0 x 10⁻¹⁰ / 1.0 x 10⁻¹²) = 10 log (100) = 20 dB\r<BR>\n" );
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document.write( "**The intensity level of a whisper is 20 dB.**\r<BR>\n" );
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document.write( "**c) Siren**\r<BR>\n" );
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document.write( "A siren at 30 meters has an intensity of 1.0 x 10⁻² W/m².  Calculating the decibel level:\r<BR>\n" );
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document.write( "β = 10 log (1.0 x 10⁻² / 1.0 x 10⁻¹²) = 10 log (10¹⁰) = 100 dB\r<BR>\n" );
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document.write( "**The intensity level of the siren is 100 dB.**\r<BR>\n" );
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document.write( "**Comparison**\r<BR>\n" );
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document.write( "To find how much louder the siren is compared to the whisper, we subtract their decibel levels:\r<BR>\n" );
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document.write( "Difference = 100 dB - 20 dB = 80 dB\r<BR>\n" );
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document.write( "**The siren sounds 80 dB louder than the whisper.**<BR>\n" );
document.write( " </SPAN>\n" );
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