document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1183394>Question 1183394</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An artifact was discovered by an archeologist, who initially estimated it to be about 2500 years old. Then, after carbon-14 analysis was completed, it was determined that the carbon-14 contained in the artifact decreased from 20 grams originally to 12 grams now, so that 60% of its original amount of carbon-14 remains. After the mathematical investigation was completed, this model was created: P(t) = 20e-0.00012t, where P(t) is the amount of carbon-14 remaining after 't' years. What conclusion should be made about the age of the artifact?<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 #813679 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=813679\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> p(t) = 20 * e^(-.00012t)<BR>\n" );
document.write( "p(0) = 20<BR>\n" );
document.write( "p(t) = .6 * 20 = 12\r<BR>\n" );
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document.write( "p(t) = e^(-.00012t) = 12<BR>\n" );
document.write( "formula becomes:<BR>\n" );
document.write( "12 = 20 * e^(-.00012t)<BR>\n" );
document.write( "divide both sides of the equation by 20 to get:<BR>\n" );
document.write( "12/20 = e^(-.00012 * t)<BR>\n" );
document.write( "take the natural log of both sides of the equation to get:<BR>\n" );
document.write( "ln(12/20) = ln(e^(-.00012 * t)<BR>\n" );
document.write( "since, in general, ln(e^x) = x * ln(e), the formula becomes:<BR>\n" );
document.write( "ln(12/20) = -.00012 * t * ln(e)<BR>\n" );
document.write( "since ln(e) = 1, the equation becomes:<BR>\n" );
document.write( "ln(12/20) = -.00012 * t<BR>\n" );
document.write( "solve for t to get:<BR>\n" );
document.write( "t = ln(12/20) / -.00012 = 4256.880198 years.\r<BR>\n" );
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document.write( "confirm by replacing t in the original equation to get:<BR>\n" );
document.write( "p(4256.880198) = 20 * e^(-.00012 * 4256.880198) = 12.<BR>\n" );
document.write( "this confirms the value of t is correct.\r<BR>\n" );
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