document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=288548>Question 288548</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The value of a particular investment follows a pattern of exponential growth. In the year 2000, you invested money in a money market account. The value of your investment t years after 2000 is given by the exponential growth model A=3000e^0.053t. When will the account be worth $5097? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #209170 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=209170\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Invest money into money market account in year 2000.<BR>\n" );
document.write( "Investment is for T years.\r<BR>\n" );
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document.write( "The formula for investment growth with continuous compounding is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F+=+P%2Ae%5E%28I%2AT%29 ALIGN=MIDDLE  ALT=\"F+=+P%2Ae%5E%28I%2AT%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "F = Future Amount.<BR>\n" );
document.write( "P = Present Amount.<BR>\n" );
document.write( "I = Annual Interest Rate<BR>\n" );
document.write( "T = Number of Years\r<BR>\n" );
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document.write( "They have given you the formula as:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+3000%2Ae%5E%280.053%2AT%29 ALIGN=MIDDLE  ALT=\"A+=+3000%2Ae%5E%280.053%2AT%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "This is the same formula, except that they have replaced F with A and they have replaced P with 3000 and they have replaced I with .053.\r<BR>\n" );
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document.write( "Apparently, the money invested in the year 2000 is $3000 and the Annual Interest Rate is .053.\r<BR>\n" );
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document.write( "e is the scientific constant of 2.718281828...\r<BR>\n" );
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document.write( "The question they are asking is when you will have $5097 in the account.\r<BR>\n" );
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document.write( "$5097 is the future amount.\r<BR>\n" );
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document.write( "replace A with $5097 and your equation becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5097+=+3000%2Ae%5E%28.053%2AT%29 ALIGN=MIDDLE  ALT=\"5097+=+3000%2Ae%5E%28.053%2AT%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Divide both sides of this equation by 3000 to get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5097+%2F+3000+=+e%5E%28.053%2AT%29 ALIGN=MIDDLE  ALT=\"5097+%2F+3000+=+e%5E%28.053%2AT%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Take the log of both sides of this equation to get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%285097%2F3000%29+=+log%28e%5E%28.053%2AT%29%29 ALIGN=MIDDLE  ALT=\"log%285097%2F3000%29+=+log%28e%5E%28.053%2AT%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Since, in general, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28x%5Ey%29 ALIGN=MIDDLE  ALT=\"log%28x%5Ey%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2Alog%28x%29 ALIGN=MIDDLE  ALT=\"y%2Alog%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, your equation becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%285097%2F3000%29+=+.053%2AT%2Alog%28e%29 ALIGN=MIDDLE  ALT=\"log%285097%2F3000%29+=+.053%2AT%2Alog%28e%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Divide both sides of this equation by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=.053%2Alog%28e%29 ALIGN=MIDDLE  ALT=\".053%2Alog%28e%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%285097%2F3000%29%2F%28.053%2Alog%28e%29%29+=+T ALIGN=MIDDLE  ALT=\"log%285097%2F3000%29%2F%28.053%2Alog%28e%29%29+=+T\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Use the log function of your calculator to find the log of (5097/3000) to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=.230193379 ALIGN=MIDDLE  ALT=\".230193379\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Use the log function of your calculator to find the log of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e ALIGN=MIDDLE  ALT=\"e\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=.434294482 ALIGN=MIDDLE  ALT=\".434294482\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Substitute these values into your equation of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%285097%2F3000%29%2F%28.053%2Alog%28e%29%29+=+T ALIGN=MIDDLE  ALT=\"log%285097%2F3000%29%2F%28.053%2Alog%28e%29%29+=+T\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=.230193379%2F%28.053%2A.434294482%29+=+T ALIGN=MIDDLE  ALT=\".230193379%2F%28.053%2A.434294482%29+=+T\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Solve for T to get:\r<BR>\n" );
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document.write( "T = 10.00075175.\r<BR>\n" );
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document.write( "That's the number of years it will take for $3000 invested in the year 2000 to reach $5079 given that the annual interest rate is .053 and continuous compounding is used.\r<BR>\n" );
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document.write( "To confirm, plug that value into your original equation to see if it is true.\r<BR>\n" );
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document.write( "Your original equation is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5097+=+3000%2Ae%5E%280.053%2AT%29 ALIGN=MIDDLE  ALT=\"5097+=+3000%2Ae%5E%280.053%2AT%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Replace T with 10.00075175 to get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5097+=+3000%2Ae%5E%280.053%2A10.00075175%29 ALIGN=MIDDLE  ALT=\"5097+=+3000%2Ae%5E%280.053%2A10.00075175%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Simplify to get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5097+=+3000%2Ae%5E%28.530039843%29 ALIGN=MIDDLE  ALT=\"5097+=+3000%2Ae%5E%28.530039843%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Simplify further to get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5097+=+3000%2A1.699 ALIGN=MIDDLE  ALT=\"5097+=+3000%2A1.699\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Simplify further to get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5097+=+5097 ALIGN=MIDDLE  ALT=\"5097+=+5097\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "The equation is true, so your answer is good.\r<BR>\n" );
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document.write( "That answer is:\r<BR>\n" );
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document.write( "The account will be worth $5097 in the year 2000 + 10.00075175 which would be the year 2010.\r<BR>\n" );
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document.write( "You start off with $3000 in the year 2000.\r<BR>\n" );
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document.write( "Sometime in the tenth year after, your account will be worth $5097.\r<BR>\n" );
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document.write( "A year by year analysis of your investment growth would look like the following:<BR>\n" );
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document.write( "year   money\r\n" );
document.write( "2000   $3000\r\n" );
document.write( "2001   $3163\r\n" );
document.write( "2002   $3335\r\n" );
document.write( "2003   $3517\r\n" );
document.write( "2004   $3708\r\n" );
document.write( "2005   $3912\r\n" );
document.write( "2006   $4123\r\n" );
document.write( "2007   $4347\r\n" );
document.write( "2008   $4584\r\n" );
document.write( "2009   $4833\r\n" );
document.write( "2010   $5096.796926\r\n" );
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document.write( "The figure in the year 2010 is equivalent to $3000*e^(.053*10).\r<BR>\n" );
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document.write( "The actual answer is $3000*e^(.053*10.00075175).   This gives you exactly $5097.\r<BR>\n" );
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document.write( "It's a little more than 10 years, but that falls in the year 2010 anyway, so the answer, given in what year it occurs, is 2010.\r<BR>\n" );
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document.write( "10.00075175 years is equivalent to:\r<BR>\n" );
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document.write( "10 years and .274570421 days if we assume 1 year = 365.242199 days days in a year.\r<BR>\n" );
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document.write( "That puts you into January 1st of the year 2010.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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