document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=112151>Question 112151</A>: <I><SPAN STYLE=\"word-break: break-all;\"> This question deals with an exact formula. In the following exercise use the approximate and exact formulas to find the doubling time. Also answer the given question.\r<BR>\n" );
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document.write( "38. A family of 100 termites invades your house and grows at a rate of 20% per week. How many termites will be in your house after 1 year?\r<BR>\n" );
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document.write( "The approximate doubling time formula (rule of 70): for a quantity growing exponentially at a rate of P% per time period, the doubling time is approximately Tdouble = 70/P. This approximation works best for small growth rates and breaks down for growth rates over about 15%. This is the exact doubling time formula: For an exponentially growing quantity with a fractional growth rate r, the doubling time is Tdouble = log10 2/log10(1 + r). For an exponentially decaying quantity, we use a negative value for r (for example, if the decay rate is 5% per year, we set r = -0.05 per year); the half-life is Thalf= - log10 2/log10(1 + r). Note that the units of time used for T and r must be the same. For example, if the fractional growth rate is 0.05 per month, then the doubling time will also be measured in months.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #81819 by </B><A HREF=/tutors/aboutme.mpl?userid=stanbon><B>stanbon(75887)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=stanbon><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=81819\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 38. A family of 100 termites invades your house and grows at a rate of 20% per week. How many termites will be in your house after 1 year? <BR>\n" );
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document.write( "The approximate doubling time formula (rule of 70): for a quantity growing exponentially at a rate of P% per time period, the doubling time is approximately Tdouble = 70/P. This approximation works best for small growth rates and breaks down for growth rates over about 15%.<BR>\n" );
document.write( "T(double) = 70/20= 3.5 per week<BR>\n" );
document.write( "T(double) = 3.5*52 = 182 per year<BR>\n" );
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document.write( "This is the exact doubling time formula: For an exponentially growing quantity with a fractional growth rate r, the doubling time is Tdouble = log10 2/log10(1 + r). For an exponentially decaying quantity, we use a negative value for r (for example, if the decay rate is 5% per year, we set r = -0.05 per year); the half-life is Thalf= - log10 2/log10(1 + r). Note that the units of time used for T and r must be the same. For example, if the fractional growth rate is 0.05 per month, then the doubling time will also be measured in months.\r<BR>\n" );
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document.write( "T(double)= log(base 10)2/log(base 10)(1+e)<BR>\n" );
document.write( "T(double) = 0.3010299957/log(1+0.20) = 3.801078... per week<BR>\n" );
document.write( "T(double) = 3.801078...*52 = 197.6927.. per year<BR>\n" );
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document.write( "Cheers,<BR>\n" );
document.write( "Stan H.<BR>\n" );
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