document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=825694>Question 825694</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 8.	Radioactive materials like uranium follow the law of uninhibited decay, which is an Exponential Model. This decay causes radiation. All radioactive substances have a specific half-life, which is the time required for half of the radioactive substance to decay. Uninhibited Radioactive Decay is given by the formula:\r<BR>\n" );
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document.write( "The half-life of thorium-229 is 7,340 years. How long will it take for a sample of this substance to decay to 20% of its original amount?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #497515 by </B><A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><B>josgarithmetic(39617)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=497515\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The exponential decay model is based on <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=I%2Ae%5E%28kt%29 ALIGN=MIDDLE  ALT=\"A=I%2Ae%5E%28kt%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >; some people might prefer a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-k ALIGN=MIDDLE  ALT=\"-k\"   DISABLED_style= \"max-width:90%; height:auto;\" >, but however the choice, consistency is needed.\r<BR>\n" );
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document.write( "A = amount at time t<BR>\n" );
document.write( "I = initial amount, for t=0<BR>\n" );
document.write( "k = a constant<BR>\n" );
document.write( "t = time in years\r<BR>\n" );
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document.write( "First, find k.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28A%2FI%29=ln%28e%5E%28kt%29%29 ALIGN=MIDDLE  ALT=\"ln%28A%2FI%29=ln%28e%5E%28kt%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28A%2FI%29=kt%2A1 ALIGN=MIDDLE  ALT=\"ln%28A%2FI%29=kt%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=%281%2Ft%29ln%28A%2FI%29 ALIGN=MIDDLE  ALT=\"k=%281%2Ft%29ln%28A%2FI%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The given halflife is for t=7340 years, and for which <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%281%2F2%29I ALIGN=MIDDLE  ALT=\"A=%281%2F2%29I\"   DISABLED_style= \"max-width:90%; height:auto;\" >;<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=%281%2F7340%29ln%281%2F2%29 ALIGN=MIDDLE  ALT=\"k=%281%2F7340%29ln%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-9.44%2A10%5E%28-5%29 ALIGN=MIDDLE  ALT=\"k=-9.44%2A10%5E%28-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Not sure if this will render properly but the model is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28A=I%2Ae%5E%28-0.0000944%2At%29%29 ALIGN=MIDDLE  ALT=\"highlight%28A=I%2Ae%5E%28-0.0000944%2At%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Your specific question will use A=0.20, and I=1.  You would then solve the model equation for t.<BR>\n" );
document.write( "Recall from, \"First, find k\", a slight change in step gives:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28t=%281%2Fk%29ln%28A%2FI%29%29 ALIGN=MIDDLE  ALT=\"highlight%28t=%281%2Fk%29ln%28A%2FI%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >; just substitute the values known. </SPAN>\n" );
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