document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=91677>Question 91677</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Ten years ago your friend invested in a plot of land with the idea of eventually building a summer cottage on it. His plan was to place a \"nest egg\" of $12,000 in a long term Certificate of Deposit which paid 8% interest over a period of 20 years, compounded monthly. At the end of the 20 years, he would use the money to build the cottage. \r<BR>\n" );
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document.write( "The problem was the plot had 8 prairie dogs on it 10 years ago when he acquired the ground.\r<BR>\n" );
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document.write( "This year he made a count of the prairie dogs and found there were 102 on the same plot! A biologist friend assured him that the prairie dogs were growing exponentially according to the law of natural growth, and something needed to be done.<BR>\n" );
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document.write( "1. Write the formula for natural exponential growth for this particular prairie dog population. <BR>\n" );
document.write( "Hint: Find the \"k\" value for this population, and put it in your formula. <BR>\n" );
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document.write( "2. Once you have this formula, predict: <BR>\n" );
document.write( "a. approximately how many years from now will the population reach 365 prairie dogs. \r<BR>\n" );
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document.write( "b. approximately how large the population will be 10 years from now. <BR>\n" );
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document.write( "3. Write the formula for compound interest and calculate how much money your friend will have on deposit (in the account described above) after 20 years (10 years from now).  <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 #66581 by </B><A HREF=/tutors/aboutme.mpl?userid=stanbon><B>stanbon(75887)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=stanbon><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=66581\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The problem was the plot had 8 prairie dogs on it 10 years ago when he acquired the ground. <BR>\n" );
document.write( "This year he made a count of the prairie dogs and found there were 102 on the same plot! A biologist friend assured him that the prairie dogs were growing exponentially according to the law of natural growth, and something needed to be done.\r<BR>\n" );
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document.write( "1. Write the formula for natural exponential growth for this particular prairie dog population. <BR>\n" );
document.write( "You have two points: (-10,8) and (0,102)<BR>\n" );
document.write( "x is years from now ; y = # of prairie dogs.<BR>\n" );
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document.write( "Exponential Form:<BR>\n" );
document.write( "y = ab^x<BR>\n" );
document.write( "Substitute the points into the form:<BR>\n" );
document.write( "1st: 8 = ab^-10<BR>\n" );
document.write( "2nd: 102 = ab^0<BR>\n" );
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document.write( "From the 2nd equation you get a = 102<BR>\n" );
document.write( "Substitute that into the 1st equation and solve for \"b\".<BR>\n" );
document.write( "8 = 102b^-10<BR>\n" );
document.write( "4/51 = b^-10<BR>\n" );
document.write( "b^10 = 5/4<BR>\n" );
document.write( "b = 1.022565<BR>\n" );
document.write( "EQUATION:<BR>\n" );
document.write( "y = 102(1.022565..)^x<BR>\n" );
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document.write( "2. Once you have this formula, predict: <BR>\n" );
document.write( "a. approximately how many years from now will the population reach 365 prairie dogs.<BR>\n" );
document.write( "365 = 102(1.022565...)^x<BR>\n" );
document.write( "(365/102) = 1.022565...^x<BR>\n" );
document.write( "x = log(365/102)/log(1.022565...)<BR>\n" );
document.write( "x = 2.5673 years<BR>\n" );
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document.write( "b. approximately how large the population will be 10 years from now.<BR>\n" );
document.write( "Let x = 10 in the equation and solve for y.<BR>\n" );
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document.write( "Ten years ago your friend invested in a plot of land with the idea of eventually building a summer cottage on it. His plan was to place a \"nest egg\" of $12,000 in a long term Certificate of Deposit which paid 8% interest over a period of 20 years, compounded monthly. At the end of the 20 years, he would use the money to build the cottage. \r<BR>\n" );
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document.write( "3. Write the formula for compound interest and calculate how much money your friend will have on deposit (in the account described above) after 20 years (10 years from now). <BR>\n" );
document.write( "A(t) = A(0)(1+r/n)^(nt)<BR>\n" );
document.write( "A(20) = 12000(1+0.08/12)*(12*20) <BR>\n" );
document.write( "A(20) = $59121,63<BR>\n" );
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document.write( "Cheers,<BR>\n" );
document.write( "Stan H. \r<BR>\n" );
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