document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1088830>Question 1088830</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 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&#8203; day, what is the size of the colony after 4 &#8203;days?  How long is it until there are 90,000 &#8203;mosquitoes?\r<BR>\n" );
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document.write( "What is the size of the colony after 4 &#8203;days?&#8195;&#8195;<BR>\n" );
document.write( "Approximately __3842____ mosquitoes.\r<BR>\n" );
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document.write( "&#8203;(Do not round until the final answer. Then round to the nearest whole number as&#8203; needed.)\r<BR>\n" );
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document.write( "How long is it until 90,000 mosquitoes are in the&#8203; colony? \r<BR>\n" );
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document.write( "________ days.&#8195;&#8195;<BR>\n" );
document.write( "&#8203;(Do not round until the final answer. Then round to the nearest tenth as&#8203; needed.) \r<BR>\n" );
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document.write( "I got the first but I need help with the second answer </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #703362 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=703362\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i believe the formula for uninhibited growth is:\r<BR>\n" );
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document.write( "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( "r is the growth 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 started with 1000 mosquitos and that rose to 1400 after 1 day.\r<BR>\n" );
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document.write( "f = 1400<BR>\n" );
document.write( "p = 1000<BR>\n" );
document.write( "r = what you want to find.<BR>\n" );
document.write( "n = 1 day\r<BR>\n" );
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document.write( "formula becomes 1400 = 1000 * e ^ (r * 1)\r<BR>\n" );
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document.write( "this simplifies to 1400 = 1000 * e ^ r\r<BR>\n" );
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document.write( "divide both sides of this equation by 1000 to get:\r<BR>\n" );
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document.write( "1.4 = e ^ r\r<BR>\n" );
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document.write( "take the natural log of both sides of this equation to get:\r<BR>\n" );
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document.write( "ln(1.4) = ln(e ^ r)\r<BR>\n" );
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document.write( "this becomes ln(1.4) = r * ln(e) which becomes ln(1.4) = r\r<BR>\n" );
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document.write( "you get r = ln(1.4) = .3364722366\r<BR>\n" );
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document.write( "the continuous growth rate is .3364722366 per day.\r<BR>\n" );
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document.write( "to confirm, replace g in the original equation to get:\r<BR>\n" );
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document.write( "1000 * e^.3364722366) = 1400\r<BR>\n" );
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document.write( "after 44 days, the size of the colony would be 1000 * e^(.3364722366 * 44) which is equal to 2,689,264,815.\r<BR>\n" );
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document.write( "that'a a lot of mosquitos.\r<BR>\n" );
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document.write( "to find out how long it would take for there to be 90,000 mosquitos, use the formula as shown below:\r<BR>\n" );
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document.write( "90,000 = 1000 * e^(.3364722366 * t)\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( "90,000 / 1000 = e^(.3364722366 * t).\r<BR>\n" );
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document.write( "take the natural log of both sides of this equation to get:\r<BR>\n" );
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document.write( "ln(90,000 / 1000) = ln(e^(.3364722366 * t) which becomes:\r<BR>\n" );
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document.write( "ln(90) = .3364722366 * x * ln(e) which becomes:\r<BR>\n" );
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document.write( "ln(90) = .3364722366 * x\r<BR>\n" );
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document.write( "solve for x to get x = ln(90) / .3364722366 = 13.37349469\r<BR>\n" );
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document.write( "the colony would grow to 90,000 in 13.37349469 days.\r<BR>\n" );
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document.write( "1000 * e^(.3364722366 * 13.37349469) = 90,000\r<BR>\n" );
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document.write( "using this formula, after 4 days, the number of mosquitos would be:\r<BR>\n" );
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document.write( "1000 * e^(.364722366 * 4) = 3841.6 which can be rounded to 3842.\r<BR>\n" );
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document.write( "you are correct there.\r<BR>\n" );
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document.write( "i believe you could also have used a discrete compounding formula instead of a continuous compounding formula to get the same answer.\r<BR>\n" );
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document.write( "1000 to 1400 in 1 day is a growth rate of .4 per day.\r<BR>\n" );
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document.write( "the formula for compound interest growth is 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 growth rate per time period<BR>\n" );
document.write( "n is the number of time periods.\r<BR>\n" );
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document.write( "the formula becomes 1400 = 1000 * (1 + r) ^ 1\r<BR>\n" );
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document.write( "solve for r to get r = 1400 / 1000 - 1 which is equal to .4\r<BR>\n" );
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document.write( "in 4 days, the population would go from 1000 to 1000 * (1.4)^4 = 3841.6 which can be rounded to 3842.\r<BR>\n" );
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document.write( "you get the same results whether you use the continuous compounding formula or the discrete compounding formula.\r<BR>\n" );
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document.write( "to solve for the number of days when the colony reaches 90,000, the discrete compound formula would become:\r<BR>\n" );
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document.write( "90,000 = 1000 * (1.4) ^ n\r<BR>\n" );
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document.write( "divide both sides of this equation by 1000 to get:\r<BR>\n" );
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document.write( "90 = (1.4) ^ n\r<BR>\n" );
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document.write( "take the log of  both sides of this equation to get:\r<BR>\n" );
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document.write( "log(90) = log((1.4) ^ n)\r<BR>\n" );
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document.write( "this is equivalentto:\r<BR>\n" );
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document.write( "log(90) = n * log(1.4)\r<BR>\n" );
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document.write( "solve for n to get n = log(90) / log(1.4) = 13.37349469\r<BR>\n" );
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document.write( "this is the same answer as was derived using the continuous compounding formula.\r<BR>\n" );
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document.write( "continuous compounding formula is f = p * e ^ (r * n)\r<BR>\n" );
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document.write( "discrete compounding formula is 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 growth rate per time period.<BR>\n" );
document.write( "n is the number of time periods.\r<BR>\n" );
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document.write( "note that the value of r is different, but everything else will be the same.\r<BR>\n" );
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document.write( "in the continuous compounding formula, r = .3364722366 per day\r<BR>\n" );
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document.write( "in the discrete compounding formula, r = .4 per day.\r<BR>\n" );
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document.write( "when you solve for the value of the exponent, you can use the natural log function or the regular log function of your calculator.\r<BR>\n" );
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document.write( "either one will get you the correct answer.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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