document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=635620>Question 635620</A>: <I><SPAN STYLE=\"word-break: break-all;\"> \"The population of a certain city was 30,000 in 1980 and 40,500 in 1990. If the formula y=Po(e^ht) applies to the growth citys population, what is the population projected to be in the year 2010?\"\r<BR>\n" );
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document.write( "not sure how to set up the information for the equation </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #400429 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=400429\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "You start with <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\ =\ P_oe^{ht}\">\r<BR>\n" );
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document.write( "You know the starting population, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE P_o\"> because that is given, and you know the time, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE t\"> because it is 30 years from 1980 to 2010, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e\"> is the constant base of the natural logs, so that is known as well.  What you don't know is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE h\">, the <b><i>rate</i></b> of exponential growth.\r<BR>\n" );
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document.write( "Fortunately, we are given another data point, specifically the population in 1990, and using this, we can solve the equation for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE h\">, when <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\ =\ 40,500\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE P_o\ =\ 30,000\">, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE t\ =\ 10\"> since it is 10 years from 1980 to 1990.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 40,500\ =\ 30,000e^{10h}\">\r<BR>\n" );
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document.write( "Divide by 30,000:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ e^{10h}\ =\ 1.35\">\r<BR>\n" );
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document.write( "Take the natural log of both sides.  Actually, you could take the log to any base, as long as it is the same base on both sides of the equation, but because the equation involves <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e\">, the base of the natural logs, the arithmetic is going to be much simpler if you take the natural log of both sides:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \ln\left(e^{10h}\right)\ =\ \ln(1.35)\">\r<BR>\n" );
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document.write( "Now use <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \log_b\left(a^n\right)\ =\ n\,\cdot\,\log_b(a)\">\r<BR>\n" );
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document.write( "to write:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 10h\,\cdot\,\ln(e)\ =\ \ln(1.35)\">\r<BR>\n" );
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document.write( "Then use <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \log_b(b)\ =\ 1\"> to write:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 10h\ =\ ln(1.35)\">\r<BR>\n" );
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document.write( "(See what I mean about simpler arithmetic?)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ h\ =\ \frac{\ln(1.35)}{10}\">\r<BR>\n" );
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document.write( "Which the calculator says is close enough to 0.03 that the difference doesn't matter.\r<BR>\n" );
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document.write( "NOW, you go back to the original function:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ P_oe^{ht}\">\r<BR>\n" );
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document.write( "Plug in the numbers\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ 30,000e^{0.03\,\cdot\,30}\">\r<BR>\n" );
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document.write( "And do the arithmetic.  I'll let you run the calculator this time.  \r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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