document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=845641>Question 845641</A>: <I><SPAN STYLE=\"word-break: break-all;\"> There are 3 parts to this questions. I have been stuck on this question for a long time. PLEASE HELP!!\r<BR>\n" );
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document.write( "A storage tank contains a radioactive element with a half-life of 6500 years. Let f(t) be the percentage of the element that remains at t years since it was placed in the tank.\r<BR>\n" );
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document.write( "Question A: Find an equation of f.<BR>\n" );
document.write( "Question B: What percentage of the element will remain after 100 years?<BR>\n" );
document.write( "Question C: What percentage of the element remains after 26,000 years? [NOTE: this result could be found using the equation of f, but we'll use a different approach to find the percentage].\r<BR>\n" );
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document.write( "PLEASE HELP ME SOLVE THIS PROBLEM!!! and PLEASE SHOW WORK SO I CAN UNDERSTAND IT!\r<BR>\n" );
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document.write( "Thanks!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #509460 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=509460\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A After one half-life the fraction of the element that remains is <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;\" > .<BR>\n" );
document.write( "After <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > half-lives the fraction of the element that remains is <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;\" > 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;\" > , which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%2A%281%2F2%29=%281%2F2%29%5E2 ALIGN=MIDDLE  ALT=\"%281%2F2%29%2A%281%2F2%29=%281%2F2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "After <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > half-lives the fraction of the element that remains is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%5E3 ALIGN=MIDDLE  ALT=\"%281%2F2%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The same idea works for any number of half-lives, even if that number is not an integer.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > years is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%2F6500 ALIGN=MIDDLE  ALT=\"t%2F6500\"   DISABLED_style= \"max-width:90%; height:auto;\" > half-lives,<BR>\n" );
document.write( "After that time, the fraction of the element that remains is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%5E%22t+%2F+6500%22 ALIGN=MIDDLE  ALT=\"%281%2F2%29%5E%22t+%2F+6500%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "As a percentage, it is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28t%29=100%2A%281%2F2%29%5E%22t+%2F+6500%22 ALIGN=MIDDLE  ALT=\"f%28t%29=100%2A%281%2F2%29%5E%22t+%2F+6500%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That is an exponential function with base <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;\" > , and that is not a fashionable base.<BR>\n" );
document.write( "Calculators have exponential functions with base <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10 ALIGN=MIDDLE  ALT=\"10\"   DISABLED_style= \"max-width:90%; height:auto;\" > and with base <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e ALIGN=MIDDLE  ALT=\"e\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The most popular base for exponential functions is the irrational number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e ALIGN=MIDDLE  ALT=\"e\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The reason for that is that calculus works better with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e ALIGN=MIDDLE  ALT=\"e\"   DISABLED_style= \"max-width:90%; height:auto;\" > as a base,<BR>\n" );
document.write( "so even if you do not intend to ever study calculus, they make you use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e ALIGN=MIDDLE  ALT=\"e\"   DISABLED_style= \"max-width:90%; height:auto;\" > rather than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10 ALIGN=MIDDLE  ALT=\"10\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "It is customary to write the function as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28t%29=100%2Ae%5E%28%22-+ln+%28+2+%29+%2A+t+%2F+6500%22%29 ALIGN=MIDDLE  ALT=\"f%28t%29=100%2Ae%5E%28%22-+ln+%28+2+%29+%2A+t+%2F+6500%22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28t%29=100%2Ae%5E%28%22-+0.693%2At+%2F+6500%22%29 ALIGN=MIDDLE  ALT=\"f%28t%29=100%2Ae%5E%28%22-+0.693%2At+%2F+6500%22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "It is really the same thing, because the natural logarithm of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%5E%22t+%2F+6500%22 ALIGN=MIDDLE  ALT=\"%281%2F2%29%5E%22t+%2F+6500%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28t%2F6500%29%2Aln%281%2F2%29=%28t%2F6500%29%2A%28-ln%282%29%29=-ln%282%29%2At%2F6500 ALIGN=MIDDLE  ALT=\"%28t%2F6500%29%2Aln%281%2F2%29=%28t%2F6500%29%2A%28-ln%282%29%29=-ln%282%29%2At%2F6500\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "People memorize the \"formula\" for the exponential function with base <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e ALIGN=MIDDLE  ALT=\"e\"   DISABLED_style= \"max-width:90%; height:auto;\" > without understanding it.<BR>\n" );
document.write( "I never memorized that formula, but I can deduce the one with <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;\" > as a base from the definition of half-life, and then I can \"translate\" it to base <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e ALIGN=MIDDLE  ALT=\"e\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "B When <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=100 ALIGN=MIDDLE  ALT=\"t=100\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.693%2At%2F6500=-69.3%2F6500 ALIGN=MIDDLE  ALT=\"-0.693%2At%2F6500=-69.3%2F6500\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28t%29=100%2Ae%5E%28%22-69.3+%2F+6500%22%29=100%2A0.989=98.9 ALIGN=MIDDLE  ALT=\"f%28t%29=100%2Ae%5E%28%22-69.3+%2F+6500%22%29=100%2A0.989=98.9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So after 100 years 98.9% of the radioactive element remains.<BR>\n" );
document.write( " <BR>\n" );
document.write( "C <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=26000%2F6500=4 ALIGN=MIDDLE  ALT=\"26000%2F6500=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > so 26,000 years is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > half-lives.<BR>\n" );
document.write( "The fraction that remains after <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > half-lives is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%5E4=1%2F16=0.0625 ALIGN=MIDDLE  ALT=\"%281%2F2%29%5E4=1%2F16=0.0625\"   DISABLED_style= \"max-width:90%; height:auto;\" > That is 6.25% .<BR>\n" );
document.write( "No formula used. We do not need <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e ALIGN=MIDDLE  ALT=\"e\"   DISABLED_style= \"max-width:90%; height:auto;\" > for that. </SPAN>\n" );
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