document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1157957>Question 1157957</A>: <I><SPAN STYLE=\"word-break: break-all;\">  The population of Plano, Texas, which follows the exponential growth model, increased from<BR>\n" );
document.write( " 222,030 in 2000 to 259,841 in 2010. (Source: U.S. Census Bureau, www.census.gov)<BR>\n" );
document.write( " Work is required!\r<BR>\n" );
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document.write( " a. Find the exponential growth rate, k. Give the exact rate. \r<BR>\n" );
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document.write( " b. Write the exponential growth function that models the population after t years.\r<BR>\n" );
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document.write( " c. What is the projected population in 2020?\r<BR>\n" );
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document.write( " d. How long should it take the population to double?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854522 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5396)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=854522\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Exponential functions are functions where the variable is in the exponent, such as  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%5Ex ALIGN=MIDDLE  ALT=\"y=2%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Exponential functions of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=A%2Ae%5Ekt ALIGN=MIDDLE  ALT=\"y=A%2Ae%5Ekt\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" > are constants<BR>\n" );
document.write( "are used to model exponential growth and exponential decay.<BR>\n" );
document.write( "Let's call our function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P ALIGN=MIDDLE  ALT=\"P\"   DISABLED_style= \"max-width:90%; height:auto;\" > for  population.<BR>\n" );
document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > be the number of years after 2000.<BR>\n" );
document.write( "The function is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P=A%2Ae%5Ekt ALIGN=MIDDLE  ALT=\"P=A%2Ae%5Ekt\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For the year 2000, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0 ALIGN=MIDDLE  ALT=\"t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P=222030 ALIGN=MIDDLE  ALT=\"P=222030\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=222030=A%2Ae%5E%28k%2A0%29 ALIGN=MIDDLE  ALT=\"222030=A%2Ae%5E%28k%2A0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=222030=A%2Ae%5E0 ALIGN=MIDDLE  ALT=\"222030=A%2Ae%5E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=222030=A%2A1%7D%7D+--%3E%7B%7B%7Bhighlight%28A=222030%29 ALIGN=MIDDLE  ALT=\"222030=A%2A1%7D%7D+--%3E%7B%7B%7Bhighlight%28A=222030%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For the year 2010, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=10 ALIGN=MIDDLE  ALT=\"t=10\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P=259841 ALIGN=MIDDLE  ALT=\"P=259841\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=259841=222030%2Ae%5E%28k%2A10%29 ALIGN=MIDDLE  ALT=\"259841=222030%2Ae%5E%28k%2A10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=259841%2F222030=e%5E%28k%2A10%29 ALIGN=MIDDLE  ALT=\"259841%2F222030=e%5E%28k%2A10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28259841%2F222030%29=k%2A10 ALIGN=MIDDLE  ALT=\"ln%28259841%2F222030%29=k%2A10\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28k=ln%28259841%2F222030%29%2F10%29 ALIGN=MIDDLE  ALT=\"highlight%28k=ln%28259841%2F222030%29%2F10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That is the exact value.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=approximately ALIGN=MIDDLE  ALT=\"k=approximately\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.0157257 ALIGN=MIDDLE  ALT=\"0.0157257\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=approximately ALIGN=MIDDLE  ALT=\"k=approximately\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.01573 ALIGN=MIDDLE  ALT=\"0.01573\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "We could write the growth function as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P=222030%2Ae%5E%28%28%22ln%28259841%2F222030%29%2F10%22%29%2At%29 ALIGN=MIDDLE  ALT=\"P=222030%2Ae%5E%28%28%22ln%28259841%2F222030%29%2F10%22%29%2At%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P=222030%2Ae%5E%28ln%28%22259841%2F222030%22%29%2A%28%22t%2F10%22%29%29 ALIGN=MIDDLE  ALT=\"P=222030%2Ae%5E%28ln%28%22259841%2F222030%22%29%2A%28%22t%2F10%22%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P=222030%2A%28e%5Eln%28%22259841%2F222030%22%29%29%5E%28%22t%2F10%22%29 ALIGN=MIDDLE  ALT=\"P=222030%2A%28e%5Eln%28%22259841%2F222030%22%29%29%5E%28%22t%2F10%22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P=222030%2A%28259841%2F222030%29%5E%28%22t%2F10%22%29 ALIGN=MIDDLE  ALT=\"P=222030%2A%28259841%2F222030%29%5E%28%22t%2F10%22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "or maybe use the approximate value for k, and write it as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P=222030%2Ae%5E%280.01573t%29 ALIGN=MIDDLE  ALT=\"P=222030%2Ae%5E%280.01573t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "We can calculate the projected population in 2010 and 2020, using the equations found above.<BR>\n" );
document.write( "For 2010, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=10 ALIGN=MIDDLE  ALT=\"t=10\"   DISABLED_style= \"max-width:90%; height:auto;\" > . Substituting that value into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P=222030%2Ae%5E%280.01573t%29 ALIGN=MIDDLE  ALT=\"P=222030%2Ae%5E%280.01573t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P=222030%2Ae%5E%280.01573%2A10%29=222030%2Ae%5E0.1573=222030%2A1.17034667=259852 ALIGN=MIDDLE  ALT=\"P=222030%2Ae%5E%280.01573%2A10%29=222030%2Ae%5E0.1573=222030%2A1.17034667=259852\"   DISABLED_style= \"max-width:90%; height:auto;\" >(rounded to whole number).<BR>\n" );
document.write( "However using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P=222030%2Ae%5E%280.0157257t%29 ALIGN=MIDDLE  ALT=\"P=222030%2Ae%5E%280.0157257t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P=222030%2Ae%5E%280.0157257%2A10%29=222030%2Ae%5E0.157257=222030%2A1.17029634=25984.08967 ALIGN=MIDDLE  ALT=\"P=222030%2Ae%5E%280.0157257%2A10%29=222030%2Ae%5E0.157257=222030%2A1.17029634=25984.08967\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=259841 ALIGN=MIDDLE  ALT=\"259841\"   DISABLED_style= \"max-width:90%; height:auto;\" >(rounded to whole number).<BR>\n" );
document.write( "To match all 6 digits in the population for 2010, we need the exact value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or at least a better approximation and we need to carry more digits through the calculations.<BR>\n" );
document.write( "For 2020 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=20 ALIGN=MIDDLE  ALT=\"t=20\"   DISABLED_style= \"max-width:90%; height:auto;\" > , substituting it into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P=222030%2Ae%5E%280.0157257t%29 ALIGN=MIDDLE  ALT=\"P=222030%2Ae%5E%280.0157257t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P=222030%2Ae%5E%220.0157257%2A20%2F10%22=222030%2Ae%5E0.03146514=222030%2A1.3695943527=304090.85 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Rounding to whole numbers, we get that the projected population in 2020 is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28304091%29 ALIGN=MIDDLE  ALT=\"highlight%28304091%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If we use the more accurate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P=222030%2A%28259841%2F222030%29%5E%22t%2F10%22 ALIGN=MIDDLE  ALT=\"P=222030%2A%28259841%2F222030%29%5E%22t%2F10%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P=222030%2A%28259841%2F222030%29%5E%2220%2F10%22=222030%2A%28259841%2F222030%29%5E2=259841%5E2%2F222030=304091.09 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Rounding to whole numbers, we get that the projected population in 2020 is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28304091%29 ALIGN=MIDDLE  ALT=\"highlight%28304091%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "Using the equation with the exact value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P=222030%2Ae%5E%28%28%22ln%28259841%2F222030%29%2F10%22%29%2At%29 ALIGN=MIDDLE  ALT=\"P=222030%2Ae%5E%28%28%22ln%28259841%2F222030%29%2F10%22%29%2At%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we could set <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P=2%2A222030=444060 ALIGN=MIDDLE  ALT=\"P=2%2A222030=444060\"   DISABLED_style= \"max-width:90%; height:auto;\" > and solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2=e%5E%28%28%22ln%28259841%2F222030%29%2F10%22%29%2At%29 ALIGN=MIDDLE  ALT=\"2=e%5E%28%28%22ln%28259841%2F222030%29%2F10%22%29%2At%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%282%29=%22ln%28259841%2F222030%29%2F10%22%2At ALIGN=MIDDLE  ALT=\"ln%282%29=%22ln%28259841%2F222030%29%2F10%22%2At\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=10%2Aln%282%29%2F%22ln%28259841%2F222030%29%22=44.077 ALIGN=MIDDLE  ALT=\"t=10%2Aln%282%29%2F%22ln%28259841%2F222030%29%22=44.077\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded to 3 decimal places).<BR>\n" );
document.write( "Rounding to whole numbers, it takes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2844%29 ALIGN=MIDDLE  ALT=\"highlight%2844%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > years for the population to double at the rate observed between 2000 and 2010. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );