document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1100500>Question 1100500</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The size of a certain insect population at time t (in days) obeys the function <BR>\n" );
document.write( "P(t)=281e^0.09t<BR>\n" );
document.write( "1. When will the population reach 1120 insects?\r<BR>\n" );
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document.write( "Round your answer to the nearest whole number.<BR>\n" );
document.write( "2.When will the insect population double?\r<BR>\n" );
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document.write( "Round your answer to the nearest whole number.\r<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 #715008 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=715008\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> p(t) = 281 * e^(.09 * t)\r<BR>\n" );
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document.write( "t is the time in days.\r<BR>\n" );
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document.write( "the population will reach 1120 insects when p(t) = 1120.\r<BR>\n" );
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document.write( "formula becomes 1120 = 281 * e^(.09 * t)\r<BR>\n" );
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document.write( "divide both sides by 281 to get:<BR>\n" );
document.write( "1120/281 = e^(.09 * t)<BR>\n" );
document.write( "take the natural log of both sides of this equaiton to get:<BR>\n" );
document.write( "ln(1120/281) = ln(e^(.09 * t))<BR>\n" );
document.write( "this becomes:<BR>\n" );
document.write( "ln(1120/281) = .09 * t * ln(e) which becomes:<BR>\n" );
document.write( "ln(1120/281) = .09 * t<BR>\n" );
document.write( "solve for t to get:<BR>\n" );
document.write( "t = ln(1120/281) / .09 = 15.36365883.\r<BR>\n" );
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document.write( "the population will reach 1120 in 15.36365883 days.\r<BR>\n" );
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document.write( "281 * e^(.09 * 15.36365883) = 1120, confirming the solution is correct.\r<BR>\n" );
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document.write( "2 basic principles involved.\r<BR>\n" );
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document.write( "ln(e^x) = x*ln(e).<BR>\n" );
document.write( "ln(e) = 1\r<BR>\n" );
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document.write( "to find out when the insect population will double, do the following.\r<BR>\n" );
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document.write( "p(t) = 281 * e^(.09 * t)\r<BR>\n" );
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document.write( "let p(t) = 2 * 281 = 562\r<BR>\n" );
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document.write( "formula becomes 562 = 281 * e^(.09 * t)\r<BR>\n" );
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document.write( "do the following:\r<BR>\n" );
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document.write( "divide both sides of the equation by 281 to get:<BR>\n" );
document.write( "562/281 = e^(.09 * t)<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "2 = e^(.09 * t)<BR>\n" );
document.write( "take the natural log of both sides of the equation to get:<BR>\n" );
document.write( "ln(2) = ln(e^(.09 * t))<BR>\n" );
document.write( "this becomes:<BR>\n" );
document.write( "ln(2) = .09 * t * ln(e) which becomes:<BR>\n" );
document.write( "ln(2) = .09 * t<BR>\n" );
document.write( "solve for t to get:<BR>\n" );
document.write( "t = ln(2) / .09 = 7.70163534\r<BR>\n" );
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document.write( "the insect population will double in 7.70163534 days.\r<BR>\n" );
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document.write( "281 * e^(.09 * 7.70163534) = 562, confirming the solution is correct, because 562 is double 281.\r<BR>\n" );
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document.write( "rounding your solutions to the nearest integer results in.\r<BR>\n" );
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document.write( "population will reach 1120 in 15 years.<BR>\n" );
document.write( "population will double in 8 years.\r<BR>\n" );
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