document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=487797>Question 487797</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help!!!<BR>\n" );
document.write( "I have this problem to solve with a detailed answer please<BR>\n" );
document.write( "A colony of bacteria can be modeled by N(t)= 1000e ^0.0014t, where N is measured in bacteria per millilter and t is minutes, evaluate N(0) and interpret results, estimate how long it takes for N to double<BR>\n" );
document.write( "Thank you please I need to have this answered for my final exam today </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #333141 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=333141\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> colony of bacteria are modeled by n(t) = 1000 * e^(0.00145 * t).<BR>\n" );
document.write( "n is measured in bacteria per milliliter.<BR>\n" );
document.write( "t is in minutes.<BR>\n" );
document.write( "evaluate n(0) and interpret results.<BR>\n" );
document.write( "estimate how long it takes for n to double.\r<BR>\n" );
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document.write( "n(0) would be equal to 1000 * e^(0.00145 * 0) which would be equal to 1000 * e^0 which would be equal to 1000 * 1 which would be equal to 1000 bacteria per milliliter.\r<BR>\n" );
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document.write( "this is because e^0 = 1.<BR>\n" );
document.write( "actually, anything to the 0 power is equal to 1 except 0.<BR>\n" );
document.write( "0^0 cannot be determined.<BR>\n" );
document.write( "that's not your problem though.<BR>\n" );
document.write( "in your problem, e^0 = 1 and the formula of:<BR>\n" );
document.write( "n(0) = 1000 * e^(0.00145 * t) becomes:<BR>\n" );
document.write( "n(0) = 1000 * e^(0.00145 * 0) which becomes:<BR>\n" );
document.write( "n(0) = 1000 * e^0 which becomes:<BR>\n" );
document.write( "n(0) = 1000 * 1 which becomes:<BR>\n" );
document.write( "n(0) = 1000<BR>\n" );
document.write( "since n is expressed in number of bacteria per milliliter then:<BR>\n" );
document.write( "n(0) = 1000 bacteria per milliliter.\r<BR>\n" );
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document.write( "your formula of:<BR>\n" );
document.write( "n(0) = 1000 * e^(0.00145 * 0) becomes:<BR>\n" );
document.write( "1000 = 1000 * e^(0.00145 * 0)<BR>\n" );
document.write( "divide both sides of this equation by 1000 and you get:<BR>\n" );
document.write( "1 = e^(0.00145*0)<BR>\n" );
document.write( "this becomes:<BR>\n" );
document.write( "1 = e^0 which becomes:<BR>\n" );
document.write( "1 = 1<BR>\n" );
document.write( "this confirms the calculation was good.\r<BR>\n" );
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document.write( "in order for the bacteria to double, it would have to go from 1000 to 2000.<BR>\n" );
document.write( "you have n(t) = 1000 * e^(0.00145 * t)<BR>\n" );
document.write( "you want to know the value of t when n(t) is equal to 2000.<BR>\n" );
document.write( "your formula becomes:<BR>\n" );
document.write( "2000 = 1000 * e^(0.00145 * t)<BR>\n" );
document.write( "divide both sides of this equation by 1000 to get:<BR>\n" );
document.write( "2 = e^(0.00145 * t)<BR>\n" );
document.write( "take the natural log of both sides of this equation to get:<BR>\n" );
document.write( "ln(2) = ln(e^(0.00145 * t))<BR>\n" );
document.write( "since, in general, log(a^x) = x * log(a), then your equation of:<BR>\n" );
document.write( "ln(2) = ln(e^(0.00145 * t)) becomes:<BR>\n" );
document.write( "ln(2) = 0.00145 * t * ln(e)<BR>\n" );
document.write( "since ln(e) is equal to 1, then your this equation becomes:<BR>\n" );
document.write( "ln(2) = 0.00145 * t<BR>\n" );
document.write( "divide both sides of this equation by 0.00145 and you get:<BR>\n" );
document.write( "ln(2) / 0.00145 = t<BR>\n" );
document.write( "use your calculator to solve for t to get:<BR>\n" );
document.write( "t = 478.l0325383<BR>\n" );
document.write( "if we did this right, it should take 478.10325383 minutes for the bacteria to double.<BR>\n" );
document.write( "we test this by substituting that value for t in the equation of:<BR>\n" );
document.write( "2000 = 1000 * e^(0.00145 * t) to get:<BR>\n" );
document.write( "2000 = 1000 * e^(0.00145 * 478.10325383) which becomes:<BR>\n" );
document.write( "2000 = 1000 * e^(.693147181) which becomes:<BR>\n" );
document.write( "2000 = 2000<BR>\n" );
document.write( "the formula works, so the answer is that:<BR>\n" );
document.write( "the bacteria will double in 478.10325383 minutes.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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