document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1118985>Question 1118985</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Using the formula for exponential population growth:  New population size = r*N + N, calculate the new population size in the data table below: \r<BR>\n" );
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document.write( "Number of generations 	Populations size (N)	Growth rate<BR>\n" );
document.write( "0	10	0.6<BR>\n" );
document.write( "1	16	0.6<BR>\n" );
document.write( "2	??	 <BR>\n" );
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document.write( "a. 22.6\r<BR>\n" );
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document.write( "b. 24.2\r<BR>\n" );
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document.write( "c.25.6\r<BR>\n" );
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document.write( "		<BR>\n" );
document.write( "d. 30.4\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 #734443 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=734443\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Generation 0: N = 10; r = 0.6<br><BR>\n" );
document.write( "Generation 1: N(new) = r*N+N = 0.6(10)+10 = 6+10 = 16; r = 0.6<br><BR>\n" );
document.write( "Generation 2: N(new) = r*N+N = 0.6(16)+16 = 9.6+16 = 25.6<br><BR>\n" );
document.write( "Answer c<br><BR>\n" );
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document.write( "I hope this problem is from an introductory lesson on exponential growth.  The recursive process used is extremely inefficient; to find the population after 20 generations you would have to perform the defined calculation 20 times.<br><BR>\n" );
document.write( "It is far more efficient to use an explicit formula for the population after n generations.  The recursive formula for the new population size,<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%2AN%2BN ALIGN=MIDDLE  ALT=\"r%2AN%2BN\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "can be written as<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N%28r%2B1%29 ALIGN=MIDDLE  ALT=\"N%28r%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N%281%2Br%29 ALIGN=MIDDLE  ALT=\"N%281%2Br%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >;<br><BR>\n" );
document.write( "then the population after n generations is simply the beginning population, multiplied by the \"growth factor\" (1+r) n times:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28n%29+=+N%281%2Br%29%5En ALIGN=MIDDLE  ALT=\"P%28n%29+=+N%281%2Br%29%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "For your problem the populations after 1 and 2 generations are then<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%281%29+=+10%281.6%29+=+16 ALIGN=MIDDLE  ALT=\"P%281%29+=+10%281.6%29+=+16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%282%29+=+10%281.6%29%5E2+=+10%282.56%29+=+25.6 ALIGN=MIDDLE  ALT=\"P%282%29+=+10%281.6%29%5E2+=+10%282.56%29+=+25.6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "To find the population after 10 generations by the recursive method would be very tedious; with this method it is a single calculation:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%2810%29+=+10%281.6%29%5E10+=+1100 ALIGN=MIDDLE  ALT=\"P%2810%29+=+10%281.6%29%5E10+=+1100\"   DISABLED_style= \"max-width:90%; height:auto;\" > (to the nearest whole number) </SPAN>\n" );
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