document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1143353>Question 1143353</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The half life of a substance is the time it takes a substance to decrease to half its initial amount. John has a pile of goo that decreases in amount at a constant rate. if John initially had 100 pounds of goo, and ten days later, he only had 25 pounds of goo, what Is  the half life of the goo? <BR>\n" );
document.write( "A)10<BR>\n" );
document.write( "B) 5<BR>\n" );
document.write( "C)7.5<BR>\n" );
document.write( "D)20 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #764139 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=764139\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> if it decays at a constant rate, then you are dealing with an equation of the form f = p * (1 + r) ^ n.\r<BR>\n" );
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document.write( "f is the future value<BR>\n" );
document.write( "p is the present value<BR>\n" );
document.write( "r is the interest rate per time period<BR>\n" );
document.write( "n is the number of time periods.\r<BR>\n" );
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document.write( "you start with:<BR>\n" );
document.write( "f = 25<BR>\n" );
document.write( "p = 100<BR>\n" );
document.write( "n = 10\r<BR>\n" );
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document.write( "the formula becomes 25 = 100 * (1 + r) ^ 10<BR>\n" );
document.write( "divide both sides of this formula by 100 to get:<BR>\n" );
document.write( ".25 = (1 + r) ^ 10<BR>\n" );
document.write( "take the 10th root of both sides of this equation to get:<BR>\n" );
document.write( ".25 ^ (1/10) = 1 + r<BR>\n" );
document.write( "subtract 1 from both sides of this equation to get:<BR>\n" );
document.write( ".25 ^ (1/10) - 1 = r<BR>\n" );
document.write( "solve for r to get:<BR>\n" );
document.write( "r = -0.129449437\r<BR>\n" );
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document.write( "confirm by replacing r in the original equation with that to get:<BR>\n" );
document.write( "25 = 100 * (1 - .129449437) ^ 10<BR>\n" );
document.write( "this results in 25 = 25, confirming the solution is correct.\r<BR>\n" );
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document.write( "to find the half life, make f = .5 and p = 1 to get:<BR>\n" );
document.write( "f = p * (1 + r) ^ n becomes .5 = (1 - .129449437) ^ n<BR>\n" );
document.write( "take the log of both sides of the equation to get:<BR>\n" );
document.write( "log(.5) = log((1 - .129449437) ^ n)<BR>\n" );
document.write( "by the properties of logarithms, this becomes:<BR>\n" );
document.write( "log(.5) = n * log(1 - .129449437)<BR>\n" );
document.write( "solve for n to get:<BR>\n" );
document.write( "n = log(.5) / log(1 - .129449437) = 5\r<BR>\n" );
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document.write( "your solution is that the half life of the goo is 5 days.\r<BR>\n" );
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document.write( "the goo deteriorates at the constant rate of 12.9449437% per day.<BR>\n" );
document.write( "that means it loses 12.9449437% of its mass every day.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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