document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=154765>Question 154765</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Still don't undertand this and I have a test tomorrow...yikes!!\r<BR>\n" );
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document.write( "Here's the question:\r<BR>\n" );
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document.write( "A picture that was supposedly painted by Vermeer (1632-1675) contains 99.5% of it's carbon=14. The half life for carbon=14 is 5730 years.  Is the picture a fake?\r<BR>\n" );
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document.write( "I know the formula to use is(first, Q_0 = Q-naught) Q=Q_0e^-k5730.  And I know Q should equal 1/2, but I can't seem to put it all together.  Please help!\r<BR>\n" );
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document.write( "Thanks,<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 #113975 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=113975\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28t%29=A%5B0%5De%5E%28kt%29 ALIGN=MIDDLE  ALT=\"A%28t%29=A%5B0%5De%5E%28kt%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the general exponential formula\r<BR>\n" );
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document.write( "Since we're talking about the half life, this means that when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=5730 ALIGN=MIDDLE  ALT=\"t=5730\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28t%29=%281%2F2%29A%5B0%5D ALIGN=MIDDLE  ALT=\"A%28t%29=%281%2F2%29A%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >. In other words, when the time is 5,730 years, the amount left over will be half of the original amount.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29A%5B0%5D=A%5B0%5De%5E%28k%2A5730%29 ALIGN=MIDDLE  ALT=\"%281%2F2%29A%5B0%5D=A%5B0%5De%5E%28k%2A5730%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28t%29=%281%2F2%29A%5B0%5D ALIGN=MIDDLE  ALT=\"A%28t%29=%281%2F2%29A%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=5730 ALIGN=MIDDLE  ALT=\"t=5730\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This equation represents the half life. We can now solve for the growth/decay constant \"k\"\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2=e%5E%285730k%29 ALIGN=MIDDLE  ALT=\"1%2F2=e%5E%285730k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5B0%5D ALIGN=MIDDLE  ALT=\"A%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%281%2F2%29=ln%28e%5E%285730k%29%29 ALIGN=MIDDLE  ALT=\"ln%281%2F2%29=ln%28e%5E%285730k%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the natural log of both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%281%2F2%29=5730k%2Aln%28e%29 ALIGN=MIDDLE  ALT=\"ln%281%2F2%29=5730k%2Aln%28e%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite the right side using the identity  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28x%5Ey%29=y%2Aln%28x%29%29 ALIGN=MIDDLE  ALT=\"ln%28x%5Ey%29=y%2Aln%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.693147=5730k%281%29 ALIGN=MIDDLE  ALT=\"-0.693147=5730k%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the natural log of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get approximately -0.693147 and take the natural log of \"e\" to get 1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.693147=5730k ALIGN=MIDDLE  ALT=\"-0.693147=5730k\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.693147%2F5730=k ALIGN=MIDDLE  ALT=\"-0.693147%2F5730=k\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 5,730 to isolate k.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.000121=k ALIGN=MIDDLE  ALT=\"-0.000121=k\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide\r<BR>\n" );
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document.write( "So the growth/decay constant is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-0.000121 ALIGN=MIDDLE  ALT=\"k=-0.000121\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28t%29=A%5B0%5De%5E%28-0.000121t%29 ALIGN=MIDDLE  ALT=\"A%28t%29=A%5B0%5De%5E%28-0.000121t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Now I'm assuming that we're using the time frame from 1632 to 1675 (and ignoring the current year 2008). So this would mean that the time span is 1675-1632=43 years. So the claim is that the painting is 43 years old (from the viewpoint of someone living in 1675).\r<BR>\n" );
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document.write( "So if the claim is correct, then if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=43 ALIGN=MIDDLE  ALT=\"t=43\"   DISABLED_style= \"max-width:90%; height:auto;\" > is plugged into the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28t%29=A%5B0%5De%5E%28-0.000121t%29 ALIGN=MIDDLE  ALT=\"A%28t%29=A%5B0%5De%5E%28-0.000121t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then the resulting amount of carbon should be 99.5% of the original amount. Let's see if this is true:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28t%29=A%5B0%5De%5E%28-0.000121t%29 ALIGN=MIDDLE  ALT=\"A%28t%29=A%5B0%5De%5E%28-0.000121t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2843%29=A%5B0%5De%5E%28-0.000121%2A43%29 ALIGN=MIDDLE  ALT=\"A%2843%29=A%5B0%5De%5E%28-0.000121%2A43%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=43 ALIGN=MIDDLE  ALT=\"t=43\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2843%29=A%5B0%5De%5E%28-0.005203%29 ALIGN=MIDDLE  ALT=\"A%2843%29=A%5B0%5De%5E%28-0.005203%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2843%29=A%5B0%5D%280.994811%29 ALIGN=MIDDLE  ALT=\"A%2843%29=A%5B0%5D%280.994811%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Raise \"e\" to the -0.005203th power to get 0.994811\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2843%29=0.994811A%5B0%5D ALIGN=MIDDLE  ALT=\"A%2843%29=0.994811A%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the terms\r<BR>\n" );
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document.write( "If we round to the nearest thousandth, we get\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.995A%5B0%5D ALIGN=MIDDLE  ALT=\"0.995A%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is 99.5% of the original amount <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5B0%5D ALIGN=MIDDLE  ALT=\"A%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that the amount left over is about 99.5% of the original amount. So this shows us that the painting is approximately 43 years old, which means that the painting was most likely created in 1632 (from the viewpoint of 1675). So there is a small chance that this picture is a fake.\r<BR>\n" );
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document.write( "Note: we introduced rounding errors due to the evaluation of logarithms and  division. So our answer is not accurate. \r<BR>\n" );
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