document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1087535>Question 1087535</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)=700e^0.05t\r<BR>\n" );
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document.write( "a)Determine the number of insects at t=0 days.\r<BR>\n" );
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document.write( "b)What is the growth rate of the insect population?\r<BR>\n" );
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document.write( "c)What is the population after 10 days?\r<BR>\n" );
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document.write( "d)When will the insect population reach 1050?\r<BR>\n" );
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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 #701859 by </B><A HREF=/tutors/aboutme.mpl?userid=mathmate><B>mathmate(429)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=mathmate><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=mathmate><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=701859\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Question:<BR>\n" );
document.write( "The size P of a certain insect population at time t (in days) obeys the function p(t)=700e^0.05t<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 1050?<BR>\n" );
document.write( "e)When will the insect population double?<BR>\n" );
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document.write( "Solution:<BR>\n" );
document.write( "Given :  P(t)=700e^(0.5t), t in days<BR>\n" );
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document.write( "(a) number of insects at t=0 days<BR>\n" );
document.write( "number of insects = P(0)=700(1)=700.<BR>\n" );
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document.write( "(b) the growth rate is the percentage increase between days<BR>\n" );
document.write( "=P(t+1)/P(t)=700e^(0.5(t+1))/700e^(0.5t)=e^(0.5(1))=e^0.5=1.64872<BR>\n" );
document.write( "therefore the growth rate is 64.872% per day.<BR>\n" );
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document.write( "(c) population after 10 days<BR>\n" );
document.write( "P(10)=700e^(0.5*10)=700(148.41)=103889<BR>\n" );
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document.write( "(d) the population will reach 1050 when P(t)=1050, or<BR>\n" );
document.write( "700e^0.5t=1050 =><BR>\n" );
document.write( "e^(0.5t)=1050/700=1.5<BR>\n" );
document.write( "take log<BR>\n" );
document.write( "0.5t=log(3/2) =><BR>\n" );
document.write( "t=2log(3/2)=0.8109 days<BR>\n" );
document.write( "=approximately 19.5 hours<BR>\n" );
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document.write( "(e) population will double when P(t)=2*700=1400<BR>\n" );
document.write( "700e^(0.5t)=1400 =><BR>\n" );
document.write( "e^(0.5t)=1400/700=2<BR>\n" );
document.write( "take log<BR>\n" );
document.write( "0.5t=log(2)<BR>\n" );
document.write( "t=2log(2)=1.3863<BR>\n" );
document.write( "=33.27 hours.<BR>\n" );
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document.write( "Note: log(x) is log to the base e, or natural log.  Log10 is mostly used in high-school. </SPAN>\n" );
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