document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=22115>Question 22115</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If the half life of some radioactive material is 8 hours, how long will it take for 420 grams of the substance to decay to 10grams? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #11755 by </B><A HREF=/tutors/aboutme.mpl?userid=rapaljer><B>rapaljer(4671)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rapaljer><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=rapaljer><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=11755\">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=+y+=+y%5B0%5D%2Ae%5E%28kt%29+ ALIGN=MIDDLE  ALT=\"+y+=+y%5B0%5D%2Ae%5E%28kt%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B0%5D=+420 ALIGN=MIDDLE  ALT=\"y%5B0%5D=+420\"   DISABLED_style= \"max-width:90%; height:auto;\" > grams.\r<BR>\n" );
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document.write( "Since the half life is 8 hours, this means that when t=8, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=+.5%2Ay%5B0%5D ALIGN=MIDDLE  ALT=\"y=+.5%2Ay%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=.5%2Ay%5B0%5D+=+y%5B0%5D%2A+e%5E%28k%2A8%29 ALIGN=MIDDLE  ALT=\".5%2Ay%5B0%5D+=+y%5B0%5D%2A+e%5E%28k%2A8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B0%5D ALIGN=MIDDLE  ALT=\"y%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28.5%2Ay%5B0%5D%29%2F%28y%5B0%5D%29+=+%28y%5B0%5D%2A+e%5E%28k%2A8%29%29%2F%28y%5B0%5D%29 ALIGN=MIDDLE  ALT=\"%28.5%2Ay%5B0%5D%29%2F%28y%5B0%5D%29+=+%28y%5B0%5D%2A+e%5E%28k%2A8%29%29%2F%28y%5B0%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=.5+=+e%5E%288k%29 ALIGN=MIDDLE  ALT=\".5+=+e%5E%288k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Solve for k, by taking the ln of each side of the equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+ln%28.5%29+=+ln%28e%5E%288k%29%29=+8k ALIGN=MIDDLE  ALT=\"+ln%28.5%29+=+ln%28e%5E%288k%29%29=+8k\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Divide both sides by 8<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28ln%28.5%29%29%2F8+=+k+ ALIGN=MIDDLE  ALT=\"+%28ln%28.5%29%29%2F8+=+k+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "k = -.0866433976 approximately (rounded to calculator accuracy).  You may want to store the number in your calculator for future use.\r<BR>\n" );
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document.write( "Now, the question is, given y= 10, how long did it take?  In other words, find t, using the value of k from the calculator--don't round it off!!\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+10+=+420+e%5E%28kt%29+ ALIGN=MIDDLE  ALT=\"+10+=+420+e%5E%28kt%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, solve for t.\r<BR>\n" );
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document.write( "Divide both sides by 420:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%2F420+=+e%5E%28kt%29+ ALIGN=MIDDLE  ALT=\"10%2F420+=+e%5E%28kt%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F42+=+e%5E%28kt%29+ ALIGN=MIDDLE  ALT=\"1%2F42+=+e%5E%28kt%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  \r<BR>\n" );
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document.write( "I strongly recommend that you NOT calculate 1/42, since if you do, you will be tempted to round off the decimal.  Do NOT round off numbers that you will use later to calculate your final answer.  Leave it in fractional form, which is EXACT!\r<BR>\n" );
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document.write( "Take the ln of each side:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%281%2F42%29+=+ln%28e%5E%28kt%29%29++=+kt+ ALIGN=MIDDLE  ALT=\"ln%281%2F42%29+=+ln%28e%5E%28kt%29%29++=+kt+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "To find t, just divide both sides by k:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28ln%281%2F42%29%29%2Fk+=+t ALIGN=MIDDLE  ALT=\"+%28ln%281%2F42%29%29%2Fk+=+t\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where k is the value that you previously calculated and may have stored in your calculator:\r<BR>\n" );
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document.write( "t= 43.1385 approximately, which rounds off to about 43 or 43.1 years.  \r<BR>\n" );
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document.write( "I did not check this, and I do make errors.  If you find an error in this, please let me know!!\r<BR>\n" );
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document.write( "R^2 at SCC\r<BR>\n" );
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document.write( " </SPAN>\n" );
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