document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1194724>Question 1194724</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Gallium-67 is used in nuclear medicine to help doctors locate inflammation and chronic infections. The patient is injected with a tracer (trace amount) that includes gallium-67, which collects in areas of inflammation and infection. The gallium-67 emits radiation that a special camera can detect. Gallium-67 has a half-life of 3.26 days.\r<BR>\n" );
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document.write( "- Give an exponential equation to represent the percentage of the original gallium-67 after t days.\r<BR>\n" );
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document.write( "- Determine the amount of gallium-67 left after 4 days.\r<BR>\n" );
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document.write( "-Solve your equation to determine the time it will take for there to be 1% of the original gallium-67.<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 #826981 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=826981\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "In my experience, scientists like to use an exponential equation using the natural base e to solve problems involving the half life of radioactive substances -- as shown by the other tutor.<br><BR>\n" );
document.write( "As a mathematician, I find it far easier to use the definition of half life directly.<br><BR>\n" );
document.write( "The half life of Gallium-67 is 3.26 days.  When time is measured in half lives, t days is t/3.26 half lives.  Since 1/2 of the original material is left after one half life, the fraction of the original amount remaining after n half lives is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%5En ALIGN=MIDDLE  ALT=\"%281%2F2%29%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "In this example, with the tracer element having a half life of 3.26 days, an equation for the fraction remaining after t days is<br><BR>\n" );
document.write( "ANSWER 1: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=A%280%29%281%2F2%29%5E%28t%2F3.26%29 ALIGN=MIDDLE  ALT=\"A=A%280%29%281%2F2%29%5E%28t%2F3.26%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (converted to a percent)<br><BR>\n" );
document.write( "Plug in t=4 days to get<br><BR>\n" );
document.write( "ANSWER 2: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%5E%284%2F3.26%29=0.4272 ALIGN=MIDDLE  ALT=\"%281%2F2%29%5E%284%2F3.26%29=0.4272\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 42.74 percent<br><BR>\n" );
document.write( "Note that result makes sense; 4 days is a bit more than one half life, so the amount remaining should be a bit less than 50%.<br><BR>\n" );
document.write( "You can find the answer to the last question by solving the equation<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%5E%28t%2F3.26%29=.01 ALIGN=MIDDLE  ALT=\"%281%2F2%29%5E%28t%2F3.26%29=.01\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "using logarithms.<br><BR>\n" );
document.write( "But finding an accurate numerical answer will require a calculator; so you might as well just use a graphing calculator to find the intersection of the graphs of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F2%29%5E%28t%2F3.26%29 ALIGN=MIDDLE  ALT=\"y=%281%2F2%29%5E%28t%2F3.26%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0.01 ALIGN=MIDDLE  ALT=\"y=0.01\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "ANSWER 3: 21.66 days<br><BR>\n" );
document.write( "Note again this result make sense also.  The amount remaining after 6 half lives would be 1/2^6=1/64; the amount remaining after 7 half lives would be 1/2^7=1/128.  1% is 1/100, so the answer should be between 6 and 7 half lives, which is roughly between 19 and 23 days.<br><BR>\n" );
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