document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171695>Question 1171695</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Coffee contains caffeine. The half-life of caffeine is 5 hours. This means the amount of caffeine in your bloodstream is reduced by 50% every 5 hours. Suppose you drink a cup of coffee that contains 320 mg of caffeine. How long will it take until there is 5 mg of caffeine left in your bloodstream? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #796607 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=796607\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "The phrasing \"is reduced by 50%\" is the same as \"reduced by half\". Every 5 hours, the amount of caffeine is reduced by half.\r<BR>\n" );
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document.write( "If you started with 320 mg of caffeine, then you'll have 320/2 = 160 mg of caffeine after 5 hours. Then you'll have 160/2 = 80 mg after another 5 hours (5+5 = 10 total so far)\r<BR>\n" );
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document.write( "We keep dividing the amount in half and keep track how much time has passed by (along with the number of half-lives)\r<BR>\n" );
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document.write( "A table is useful for this sort of thing.<BR>\n" );
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document.write( "<tr><TD >Half-life </TD><TD >Time</TD><TD >Amount</TD></tr><tr><TD >0</TD><TD >0</TD><TD >320</TD></tr><tr><TD >1</TD><TD >5</TD><TD >160</TD></tr><tr><TD >2</TD><TD >10</TD><TD >80</TD></tr><tr><TD >3</TD><TD >15</TD><TD >40</TD></tr><tr><TD >4</TD><TD >20</TD><TD >20</TD></tr><tr><TD >5</TD><TD >25</TD><TD >10</TD></tr><tr><TD >6</TD><TD >30</TD><TD >5</TD></tr>\n" );
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document.write( "The time value is in hours, and the amount represents the amount of caffeine in mg.\r<BR>\n" );
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document.write( "The table shows at time 30 hours, there's exactly 5 mg of caffeine left in the bloodstream.<BR>\n" );
document.write( "This happens after 6 half-lives have occurred.\r<BR>\n" );
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document.write( "An algebraic approach\r<BR>\n" );
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document.write( "x = number of half-lives that pass by <BR>\n" );
document.write( "y = amount of caffeine left over in the bloodstream\r<BR>\n" );
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document.write( "The equation tying together x and y is<BR>\n" );
document.write( "y = 320*(0.5)^x<BR>\n" );
document.write( "where 1/2 = 0.5 represents the multiplier indicating a 50% reduction\r<BR>\n" );
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document.write( "We could guess-and-check by plugging in random values of x to see if we get y = 5.<BR>\n" );
document.write( "But let's solve for x using algebra\r<BR>\n" );
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document.write( "Plug in y = 5 and use logs to isolate the exponent<BR>\n" );
document.write( "y = 320*(0.5)^x<BR>\n" );
document.write( "5 = 320*(0.5)^x<BR>\n" );
document.write( "5/320 = (0.5)^x<BR>\n" );
document.write( "0.015625 = (0.5)^x<BR>\n" );
document.write( "log(0.015625) = log( (0.5)^x )<BR>\n" );
document.write( "log(0.015625) = x*log( 0.5 )<BR>\n" );
document.write( "x*log( 0.5 ) = log(0.015625)<BR>\n" );
document.write( "x = log(0.015625)/log(0.5)<BR>\n" );
document.write( "x = 6<BR>\n" );
document.write( "This helps confirm the half-life we got earlier with the table.\r<BR>\n" );
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document.write( "Another confirmation method is to plug x = 6 into the equation<BR>\n" );
document.write( "y = 320*(0.5)^x<BR>\n" );
document.write( "y = 320*(0.5)^6<BR>\n" );
document.write( "y = 320*0.015625<BR>\n" );
document.write( "y = 5\r<BR>\n" );
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document.write( "x = 6 half-lives occurring mean 5x = 5*6 = 30 hours have passed by (since each half-life is 5 hours).\r<BR>\n" );
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document.write( "Answer: 30 hours<BR>\n" );
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