document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=84669>Question 84669</A>: <I><SPAN STYLE=\"word-break: break-all;\"> PLEASE PLEASE. I'M IN DESPERATE NEED OF HELP!\r<BR>\n" );
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document.write( "A mosquito population is growing at a rate of 8% each day and the current population is estimated at 200,000. Use y=y sub 0 times e^0.08t as the equation for the mosquito population.\r<BR>\n" );
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document.write( "a) how many mosquitoes are there expected to be in 10 days?\r<BR>\n" );
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document.write( "b) how many days will it take for the mosquito population to reach one million?<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 #60986 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=60986\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+y%5Bo%5D%2Ae%5E%280.08%2At%29 ALIGN=MIDDLE  ALT=\"y+=+y%5Bo%5D%2Ae%5E%280.08%2At%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "At time t = 0, the mosquito population represented in this equation by y is 200,000.<BR>\n" );
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document.write( "t represents the number of days since day zero when t = 0<BR>\n" );
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document.write( "Note that on day zero (when t = 0) the part of the equation represented by:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%280.08%2At%29 ALIGN=MIDDLE  ALT=\"e%5E%280.08%2At%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "becomes<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%280.08%2A0%29+=+e%5E0+=+1 ALIGN=MIDDLE  ALT=\"e%5E%280.08%2A0%29+=+e%5E0+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > because any number to the 0 power is 1 by definition.<BR>\n" );
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document.write( "Since this is 1 the equation is reduced to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+y%5Bo%5D ALIGN=MIDDLE  ALT=\"y+=+y%5Bo%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and since the mosquito population<BR>\n" );
document.write( "is known to be 200,000 at this time, we know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=200000+=+y%5Bo%5D ALIGN=MIDDLE  ALT=\"200000+=+y%5Bo%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So we can substitute<BR>\n" );
document.write( "200,000 for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5Bo%5D ALIGN=MIDDLE  ALT=\"y%5Bo%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28200000%29%2Ae%5E%280.08%2At%29 ALIGN=MIDDLE  ALT=\"y+=+%28200000%29%2Ae%5E%280.08%2At%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now to the problem.  In ten days (when t = 10) the population of mosquitoes becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28200000%29%2Ae%5E%280.08%2A10%29+=+%28200000%29%2Ae%5E%280.8%29 ALIGN=MIDDLE  ALT=\"y+=+%28200000%29%2Ae%5E%280.08%2A10%29+=+%28200000%29%2Ae%5E%280.8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Calculator time. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E0.8+=+2.225540928 ALIGN=MIDDLE  ALT=\"e%5E0.8+=+2.225540928\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Substitute this into the equation and it becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28200000%29%2A%282.225540928%29=445108.1857 ALIGN=MIDDLE  ALT=\"y+=+%28200000%29%2A%282.225540928%29=445108.1857\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Rounding to the nearest whole number, the equation tells us that in 10 days there will<BR>\n" );
document.write( "be 445108 mosquitoes.\r<BR>\n" );
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document.write( "To find out how many days it will take for the mosquito population to reach a million, set <BR>\n" );
document.write( "y equal to 1,000,000 in the equation and solve for t. The equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1000000+=+200000%2Ae%5E%280.08%2At%29 ALIGN=MIDDLE  ALT=\"1000000+=+200000%2Ae%5E%280.08%2At%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Simplify by dividing both sides by 200,000 to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5+=+e%5E%280.08%2At%29 ALIGN=MIDDLE  ALT=\"5+=+e%5E%280.08%2At%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Take the natural log of both sides and you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%285%29+=+ln%28e%5E%280.08%2At%29%29 ALIGN=MIDDLE  ALT=\"ln%285%29+=+ln%28e%5E%280.08%2At%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The ln of 5 is (by calculator) 1.609437912. Substitute this:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.609437912+=+ln%28e%5E%280.08%2At%29%29 ALIGN=MIDDLE  ALT=\"1.609437912+=+ln%28e%5E%280.08%2At%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "And by the power rule of logarithms, in the log of a quantity that is raised to an exponent,<BR>\n" );
document.write( "you can bring the exponent out and make it the multiplier of the log of the quantity that<BR>\n" );
document.write( "remains. In this case that means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28e%5E%280.08%2At%29%29+=+0.08%2At%2Aln%28e%29 ALIGN=MIDDLE  ALT=\"ln%28e%5E%280.08%2At%29%29+=+0.08%2At%2Aln%28e%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Substitute the right<BR>\n" );
document.write( "side into the equation in place of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28e%5E%280.08%2At%29%29 ALIGN=MIDDLE  ALT=\"ln%28e%5E%280.08%2At%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the result is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.609437912+=+0.08%2At%2Aln%28e%29%29 ALIGN=MIDDLE  ALT=\"1.609437912+=+0.08%2At%2Aln%28e%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "But ln(e) = 1. Substitute this and the equation reduces to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.609437912+=+0.08%2At%29 ALIGN=MIDDLE  ALT=\"1.609437912+=+0.08%2At%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Solve for t by dividing both sides of this equation by 0.08 and the answer becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+1.609437912%2F0.08+=+20.11797391 ALIGN=MIDDLE  ALT=\"t+=+1.609437912%2F0.08+=+20.11797391\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So it will take a little more than 20 days for the colony of mosquitoes to be more than<BR>\n" );
document.write( "one million in population.  The population should slightly exceed a million in 20 days and<BR>\n" );
document.write( "3 hours.<BR>\n" );
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document.write( "Hope this helps you to understand the problem. </SPAN>\n" );
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