document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1162793>Question 1162793</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The size P of a certain insect population at time t​ (in days) obeys the function P(t)=900e^0.03t <BR>\n" );
document.write( "​(a) Determine the number of insects at t=0 days.<BR>\n" );
document.write( "​(b) What is the growth rate of the insect​ population?<BR>\n" );
document.write( "​(c) What is the population after 10​ days?<BR>\n" );
document.write( "​(d) When will the insect population reach 1440?<BR>\n" );
document.write( "​(e) When will the insect population​ double? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #786657 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=786657\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> p(t) = 900 * e^(.03*t)<BR>\n" );
document.write( "p(t) = number of insects at t days.<BR>\n" );
document.write( "t = number of days.\r<BR>\n" );
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document.write( "at t = 0, the population is equal to 900 * e^(.03*0) = 900 * e^0 = 900 * 1 = 900.\r<BR>\n" );
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document.write( "at t = 10, the population is equal to 900 * e^(.03*10) = 900 * e^(.3) = 1214.872927.<BR>\n" );
document.write( "since you can't have part of an ant, round the answer to the nearest integer = 1215.\r<BR>\n" );
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document.write( "the formula for the number of days when the population will be equal to 1414 is 1440 = 900 * e^(.03*t)<BR>\n" );
document.write( "divide both sides of this equation by 900 to get:<BR>\n" );
document.write( "1440/900 = e^(.03*t)<BR>\n" );
document.write( "take the natural log of both sides of the equation to get ln(1440/900) = ln(e^(.03*t))<BR>\n" );
document.write( "since ln(e^x) = x*ln(e), this becomes ln(1440/900) = .03*t * ln(e)<BR>\n" );
document.write( "since ln(e) = 1, this becomes ln(1440/900) = .03*t<BR>\n" );
document.write( "solve for t to get t = ln(1440/900) / .03 = 15.66678764 days.\r<BR>\n" );
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document.write( "to confirm this is correct, replace t with 15.0594361 in the original equation to get p(t) = 900 * e^(.03 * 15.66678764) = 1440.\r<BR>\n" );
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document.write( "the insect population will double when it becomes 1800.\r<BR>\n" );
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document.write( "the formula for the number of days when the population will be equal to 1800 is 1800 = 900 * e^(.03*t)<BR>\n" );
document.write( "divide both sides of this equation by 900 to get:<BR>\n" );
document.write( "1800/900 = e^(.03*t)<BR>\n" );
document.write( "take the natural log of both sides of the equation to get ln(1800/900) = ln(e^(.03*t))<BR>\n" );
document.write( "since ln(e^x) = x*ln(e), this becomes ln(1800/900) = .03*t * ln(e)<BR>\n" );
document.write( "since ln(e) = 1, this becomes ln(1800/900) = .03*t<BR>\n" );
document.write( "solve for t to get t = ln(1800/900) / .03 = 23.10490602 days.<BR>\n" );
document.write( "to confirm this is correct, replace t with 23.10490602 in the original equation to get p(t) = 900 * e^(.03 * 23.10490602) = 1800.\r<BR>\n" );
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document.write( "to find the daily growth rate, do the following.<BR>\n" );
document.write( "start with 900.<BR>\n" );
document.write( "multiply it by e^(.03 * 1 to get 927.4090806.<BR>\n" );
document.write( "divide that by 900 to get 1.030454534.<BR>\n" );
document.write( "subtract 1 from that to get .030454534 = 3.0454534%.<BR>\n" );
document.write( "that's the daily growth rate.\r<BR>\n" );
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document.write( "to confirm, use that in the discrete growth rate formula of f = p * ( 1 + r) ^ n<BR>\n" );
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( "for example:<BR>\n" );
document.write( "the population in 10 days will be equal to 900 * (1 + .030454534) ^ 10 = 1214.872927.<BR>\n" );
document.write( "this is the same result we got before with p(t) = 900 * e^(.03 * 10)<BR>\n" );
document.write( "p(t) means the same as f.<BR>\n" );
document.write( "they both represent the future value of the insect population.<BR>\n" );
document.write( "p(t) and f tell you what the future value is.<BR>\n" );
document.write( "p tells you what the present value is.\r<BR>\n" );
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document.write( "the discrete growth rate formula of f = p * (1 + .030454534) ^ n will give you the same answer as the continuous growth rate formula of p(t) = p * e^(.03 * t)\r<BR>\n" );
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document.write( " </SPAN>\n" );
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