document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=426300>Question 426300</A>: <I><SPAN STYLE=\"word-break: break-all;\"> R(t) = 1150/0.5+22.5(2.2)^-0.065t<BR>\n" );
document.write( "                  the exponent is -0.065t\r<BR>\n" );
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document.write( "t = is the number of weeks after the research team first introduced the rabbits into the forest.\r<BR>\n" );
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document.write( "I figured out the below answers and they are right, I cannot figure out this, the rabbit population approaches ________ as time goes on?\r<BR>\n" );
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document.write( "the research team brought 50 rabbits to the forest (t=0)<BR>\n" );
document.write( "how many rabbits can be expected after 10weeks = 82 rabbits (t=10) <BR>\n" );
document.write( "how many rabbits can be expected after the first year = 557 rabbits (t=52)<BR>\n" );
document.write( "how many rabbits can be expected after 4 years = 2298 rabbits (t=208)<BR>\n" );
document.write( "how many rabbits can be expected after 5 years = 2300 rabbits (t=260)\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #296574 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3759)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=296574\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I assume the function is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%28t%29+=+1150%2F%280.5%2B22.5%282.2%29%5E%28-0.065t%29%29 ALIGN=MIDDLE  ALT=\"R%28t%29+=+1150%2F%280.5%2B22.5%282.2%29%5E%28-0.065t%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "To answer the question about the rabbit population approaching some number \"as time goes on\" it means, in mathematical terms, as t approaches infinity.<br><BR>\n" );
document.write( "To answer this question we use some logic and some knowledge about how exponents and fractions work:<ul><li>As t approaches infinity (i.e a very, very large number), the exponent, -0.065t, will be become a very, very large negative number.</li><li>Since negative exponents mean reciprocals, 2.2 to a very large negative power is a fraction: 1 over 2.2 to a very, very large positive exponent.</li><li>As t gets larger and larger, the denominator of this fraction gets larger and larger.</li><li>As the denominator of a fraction gets larger and larger (without a change in the numerator), the fraction gets smaller and smaller. In fact it gets closer and closer to zero.</li><li>So as t gets larger and larger, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2.2%5E%28-0.065t%29 ALIGN=MIDDLE  ALT=\"2.2%5E%28-0.065t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > gets closer and closer to zero.</li><li>As <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2.2%5E%28-0.065t%29 ALIGN=MIDDLE  ALT=\"2.2%5E%28-0.065t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > gets closer to zero, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=22.5%2A2.2%5E%28-0.065t%29 ALIGN=MIDDLE  ALT=\"22.5%2A2.2%5E%28-0.065t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > also gets closer and closer to zero.</li><li>As <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=22.5%2A2.2%5E%28-0.065t%29 ALIGN=MIDDLE  ALT=\"22.5%2A2.2%5E%28-0.065t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > gets closer and closer to zero, the denominator, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.5+%2B+22.5%2A2.2%5E%28-0.065t%29 ALIGN=MIDDLE  ALT=\"0.5+%2B+22.5%2A2.2%5E%28-0.065t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, gets closer and closer to 0.5.</li><li>As the denominator gets closer and closer to 0.5, R(t) gets closer and closer to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1150%2F0.5 ALIGN=MIDDLE  ALT=\"1150%2F0.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ></li><li>And since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1150%2F0.5+=+2300 ALIGN=MIDDLE  ALT=\"1150%2F0.5+=+2300\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the population of rabbits will approach 2300 \"as time goes by\".</li></ul><BR>\n" );
document.write( "P.S. R(t) only approaches 2300 for large values of t. It will never actually be equal to 2300. When you got an answer of 2300 for the population after 5 years you must have rounded off your answer to get 2300 (or you made an error.) </SPAN>\n" );
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