document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192444>Question 1192444</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Researcher are recording how much of an experimental medication is in a person’s bloodstream every hour. They discover that half-life of the medication is about 6 hours. <BR>\n" );
document.write( "Write an equation to model how much medication will be in the bloodstream after an unknown number of days for an initial dose of 𝑎.? <BR>\n" );
document.write( "Calculate how much medication is in a person’s bloodstream after 4 days if they took an initial dose of 500mg?<BR>\n" );
document.write( " How much more medication will be in a person’s bloodstream if their initial dose was 750mg?                             </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #824337 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=824337\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> if the initial dose is a, then the half life will be 1/2 * a.<BR>\n" );
document.write( "the formula for half life becomes:<BR>\n" );
document.write( "1/2 * a = a * g ^ 6<BR>\n" );
document.write( "g is the growth rate.<BR>\n" );
document.write( "divide both sides of the equation by a to get:<BR>\n" );
document.write( "1/2 * a / a = g ^ 6<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "1/2 = g ^ 6<BR>\n" );
document.write( "solve for g to get:<BR>\n" );
document.write( "g = (1/2) ^ (1/6) = .8908987181.<BR>\n" );
document.write( "that's the growth rate per hour.<BR>\n" );
document.write( "in 6 hours, a * .8908987181 ^ 6 = .5 * a.<BR>\n" );
document.write( "in other words, the life of the mediation in your bloodstream is now half of what it was when the medicine was first administered.\r<BR>\n" );
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document.write( "in terms of days, the equation becomes:<BR>\n" );
document.write( "y = a * g ^ (24 * d).<BR>\n" );
document.write( "this tells you how much life is left after d days.\r<BR>\n" );
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document.write( "to see if this formula is correct, look for the half life of 6 hours in terms of days.<BR>\n" );
document.write( "since 6 hours is .25 days, you get:<BR>\n" );
document.write( "y = a * g ^ (24 * .25) which becomes:<BR>\n" );
document.write( "y = a * g ^ 6.<BR>\n" );
document.write( "since g is equal to .8908987181, this formula becomes:<BR>\n" );
document.write( "y = a * .8908987181 ^ 6 = .5a.\r<BR>\n" );
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document.write( "the formula translated to days is:<BR>\n" );
document.write( "y = a * g ^ (24 * d), where d is the number of days.\r<BR>\n" );
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document.write( "with an initial dose of 500 mg, the formula becomes:<BR>\n" );
document.write( "y = 500 * g ^ (24 * 4) after 4 days.<BR>\n" );
document.write( "simplify to get y = 500 * g ^ 96 = .0076293945.\r<BR>\n" );
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document.write( "with an initial dose of 750 mg, the formula becomes:<BR>\n" );
document.write( "y = 750 * g ^ (24 * 4) after 4 days.<BR>\n" );
document.write( "simplify to get y = 750 * g ^ 96 = .0114440918.\r<BR>\n" );
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document.write( "the amount of medication will be .0114440918 / .0076293945 = 1.5 times as much if they started with 750 mg rather than 500 mg.\r<BR>\n" );
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document.write( "you can figure this out from the equation.\r<BR>\n" );
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document.write( "with 500 mg start, the equation becomes 500 * g ^ (24 * 4)<BR>\n" );
document.write( "with 750 mg start, the equation becomes 750 * g ^ (24 * 4)<BR>\n" );
document.write( "(500 * g ^ (24 * 4)) / (750 * g ^ (24 * 4)) becomes 750 / 500 after g ^ (24 * 4) in the numerator and denominator cancel out.<BR>\n" );
document.write( "750/500 = 1.5\r<BR>\n" );
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document.write( "the growth rate itself is g which is per hour.\r<BR>\n" );
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document.write( "you could have solved for the growth rate per days is you did the following.<BR>\n" );
document.write( "you know the half life is in 6 hours.<BR>\n" );
document.write( "6 hours is 6/24 = .25 days.<BR>\n" );
document.write( "the formula for half life becomes 1/2 = g ^ .25<BR>\n" );
document.write( "solve for g to get:<BR>\n" );
document.write( "g = (1/2) ^ (1/.25) = (1/2) ^ 4 = .0625.<BR>\n" );
document.write( "this g is the growth rate per day.\r<BR>\n" );
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document.write( "half life in number of days becomes 1/2 = g ^ .25 = .0625 ^ .25 = .5.<BR>\n" );
document.write( "you now have the half life in terms of days rather than hours.<BR>\n" );
document.write( "the advantage of doing it this way is that the rest of the problem is in days, rather than hours.\r<BR>\n" );
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document.write( "in 4 days, 500 mg becomes 500 * .0625 ^ 4 = .0076293945.<BR>\n" );
document.write( "in 4 days, 750 mg becomes 750 * .0625 ^ 4 = .0114440918.\r<BR>\n" );
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document.write( "these are the same answers you got before when the growth rate was per hour.<BR>\n" );
document.write( "it winds up being cleaner when you solve for the growth rate per day rather than per hour.\r<BR>\n" );
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document.write( "either way gets you the same answer.<BR>\n" );
document.write( " </SPAN>\n" );
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