document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1140192>Question 1140192</A>: <I><SPAN STYLE=\"word-break: break-all;\">  Question Help <BR>\n" );
document.write( "The population of a colony of mosquitoes obeys the law of uninhibited growth.  If there are 1000 mosquitoes initially and there are 1400 after 1​ day, what is the size of the colony after <BR>\n" );
document.write( "3 ​days?  How long is it until there are 70,000 ​mosquitoes? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #760691 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=760691\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> population growth or decay uses the continuous compounding formula.\r<BR>\n" );
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document.write( "that formula is f = p * e ^ (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( "e is the scientific constant of 2.718281828.....<BR>\n" );
document.write( "n is the number of time periods\r<BR>\n" );
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document.write( "these are names i assigned to the variables.<BR>\n" );
document.write( "other people might assign different names, but the formula is the same.\r<BR>\n" );
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document.write( "you are given that there are 1000 mosquitoes initially and there are 1400 after 1 day.\r<BR>\n" );
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document.write( "the formula becomes 1400 = 1000 * e ^ (r * 1)\r<BR>\n" );
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document.write( "divide both sides of the formula by 1000 to get:\r<BR>\n" );
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document.write( "1400 / 1000 = e ^ (r * 1)\r<BR>\n" );
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document.write( "take the natural log of both sides of the formula to get:\r<BR>\n" );
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document.write( "ln(1400 / 1000) = ln(e ^ (r * 1))\r<BR>\n" );
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document.write( "since ln(e ^ (r * 1)) is equal to r+ * 1 * ln(e) and since ln(e) is equal to 1, then the formula becomes:\r<BR>\n" );
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document.write( "ln(1400 / 1000) = r.\r<BR>\n" );
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document.write( "solve for r to get r = .3364722366\r<BR>\n" );
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document.write( "replace r in the original equation to confirm the solution is good.\r<BR>\n" );
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document.write( "1400 = 1000 * e ^ (r * 1) becomes 1400 = 1000 * e ^ (.3364722366 * 1) which becomes 1400 = 1400, confirming the solution is correct.\r<BR>\n" );
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document.write( "after 3 days, the size of the colony will be based on f = 1000 * e ^ (.3364722366 * 3) which becomes f = 2744\r<BR>\n" );
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document.write( "to find out how long until there are 70,000 mosquitoes, the formula becomes 70,000 = 1000 * e ^ (.3364722366 * n)\r<BR>\n" );
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document.write( "divide both sides of the equation by 1000 to get:\r<BR>\n" );
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document.write( "70,000 / 1000 = e ^ (.3364722366 * n)\r<BR>\n" );
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document.write( "take the natural log of both sides of the equation to get:\r<BR>\n" );
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document.write( "ln(70,000 / 1000) = ln(e ^ (.3364722366 * n))\r<BR>\n" );
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document.write( "since ln(e ^ (.3364722366 * n)) = .3364722366 * n * ln(e) and since ln(e) = 1, the equation becomes:\r<BR>\n" );
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document.write( "ln(70,000 / 1000) = .3364722366 * n\r<BR>\n" );
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document.write( "divide both sides of this equation by .3364722366 to get:\r<BR>\n" );
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document.write( "ln(70,000 / 1000) / .3364722366 = n\r<BR>\n" );
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document.write( "solve for n  to get n = 12.62658484 days.\r<BR>\n" );
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document.write( "replace n in the original equation to confirm this is true.\r<BR>\n" );
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document.write( "70,000 = 1000 * e ^ (.3364722366 * n) becomes 70,000 = 1000 * e ^ (.3364722366 * 12.62658484) which becomes 70,000 = 70,000, confirming the solution is correct.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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