document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=731574>Question 731574</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose you deposit $500 in an account with an annual interest rate of 8% compounded monthly.\r<BR>\n" );
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document.write( "a. Find an equation that gives the amount of money in the account after t years.<BR>\n" );
document.write( "b. Find the amount of money in the account after 5 years.<BR>\n" );
document.write( "c. How many years will it take for the account to contain $1000?<BR>\n" );
document.write( "d. If the interest were compounded continuously, how much money would the account contain after 5 years?\r<BR>\n" );
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document.write( "Thank You So Much! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #447240 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</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=447240\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Interest compounded monthly:<BR>\n" );
document.write( "The interest for 1 month would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F12 ALIGN=MIDDLE  ALT=\"1%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > of 8%, meaning <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F12%29%288%2F100%29+=2%2F300 ALIGN=MIDDLE  ALT=\"%281%2F12%29%288%2F100%29+=2%2F300\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the initial balance that month.<BR>\n" );
document.write( "So at the end of the first month you would have a balance (in $) of<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=500%2B500%282%2F300%29=500%2B500%2A2%2F300=500%281%2B2%2F300%29 ALIGN=MIDDLE  ALT=\"500%2B500%282%2F300%29=500%2B500%2A2%2F300=500%281%2B2%2F300%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the new balance would be the initial balance times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2B2%2F300%29 ALIGN=MIDDLE  ALT=\"%281%2B2%2F300%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "During the second month, you would be earning interest on the whole of that $<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=500%281%2B2%2F300%29 ALIGN=MIDDLE  ALT=\"500%281%2B2%2F300%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and that balance would be multiplied times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2B2%2F300%29 ALIGN=MIDDLE  ALT=\"%281%2B2%2F300%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get the new balance of<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=500%281%2B2%2F300%29%281%2B2%2F300%29=500%2A%281%2B2%2F300%29%5E2 ALIGN=MIDDLE  ALT=\"500%281%2B2%2F300%29%281%2B2%2F300%29=500%2A%281%2B2%2F300%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "After <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > months you would have a total of<BR>\n" );
document.write( "$<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=500%2A%281%2B2%2F300%29%5En ALIGN=MIDDLE  ALT=\"500%2A%281%2B2%2F300%29%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "a. At the end of 5 years you would have accumulated interest during <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2A%2812months%29=60months ALIGN=MIDDLE  ALT=\"5%2A%2812months%29=60months\"   DISABLED_style= \"max-width:90%; height:auto;\" > and would have a total of<BR>\n" );
document.write( "$<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=500%2A%281%2B2%2F300%29%5E60 ALIGN=MIDDLE  ALT=\"500%2A%281%2B2%2F300%29%5E60\"   DISABLED_style= \"max-width:90%; height:auto;\" >= $<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=774.92 ALIGN=MIDDLE  ALT=\"774.92\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded)<BR>\n" );
document.write( " <BR>\n" );
document.write( "b. To get to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=500%2A%281%2B2%2F300%29%5En=1000 ALIGN=MIDDLE  ALT=\"500%2A%281%2B2%2F300%29%5En=1000\"   DISABLED_style= \"max-width:90%; height:auto;\" > you will need <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > months, and we can find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > like this:<BR>\n" );
document.write( "From<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=500%2A%281%2B2%2F300%29%5En=1000 ALIGN=MIDDLE  ALT=\"500%2A%281%2B2%2F300%29%5En=1000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "taking logarithms on both sides, we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%28500%2A%281%2B2%2F300%29%5En%29%29=log%28%281000%29%29 ALIGN=MIDDLE  ALT=\"log%28%28500%2A%281%2B2%2F300%29%5En%29%29=log%28%281000%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%28500%29%29%2Bn%2Alog%28%281%2B2%2F300%29%29=3 ALIGN=MIDDLE  ALT=\"log%28%28500%29%29%2Bn%2Alog%28%281%2B2%2F300%29%29=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=%283-log%28%28500%29%29%29%2Flog%28%281%2B2%2F300%29%29 ALIGN=MIDDLE  ALT=\"n=%283-log%28%28500%29%29%29%2Flog%28%281%2B2%2F300%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That calculates as about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=104.3 ALIGN=MIDDLE  ALT=\"104.3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded)<BR>\n" );
document.write( "If you need $1000, you will have to wait for 105 months because<BR>\n" );
document.write( "after 104 months you will have<BR>\n" );
document.write( "$<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=500%2A%281%2B2%2F300%29%5E104 ALIGN=MIDDLE  ALT=\"500%2A%281%2B2%2F300%29%5E104\"   DISABLED_style= \"max-width:90%; height:auto;\" >= $<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=997.89 ALIGN=MIDDLE  ALT=\"997.89\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded)<BR>\n" );
document.write( "but after 105 months you will have<BR>\n" );
document.write( "$<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=500%2A%281%2B2%2F300%29%5E105 ALIGN=MIDDLE  ALT=\"500%2A%281%2B2%2F300%29%5E105\"   DISABLED_style= \"max-width:90%; height:auto;\" >= $<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1004.54 ALIGN=MIDDLE  ALT=\"1004.54\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded)<BR>\n" );
document.write( " <BR>\n" );
document.write( "c. Interest compounded monthly:<BR>\n" );
document.write( "If the interest was calculated and added to the balance after shorter and shorter periods, you would gain a little bit more as the periods were shortened, getting as close as you want, but never going over a certain limit.<BR>\n" );
document.write( "That limit is what they call continuous compounding.<BR>\n" );
document.write( "The balance with continuous compounding is given by the function<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=balance ALIGN=MIDDLE  ALT=\"balance\"   DISABLED_style= \"max-width:90%; height:auto;\" >= $<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=500%2Ae%5E%280.08%2Ay%29 ALIGN=MIDDLE  ALT=\"500%2Ae%5E%280.08%2Ay%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >= number of years.<BR>\n" );
document.write( "In that expression,<BR>\n" );
document.write( "$<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=500 ALIGN=MIDDLE  ALT=\"500\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the initial deposit,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.08 ALIGN=MIDDLE  ALT=\"0.08\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the 8% interest rate expresed as a decimal,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e ALIGN=MIDDLE  ALT=\"e\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an irrational number, like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5 ALIGN=MIDDLE  ALT=\"y=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get the balance after 5 years as<BR>\n" );
document.write( "$<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=500%2Ae%5E%280.08%2A5%29 ALIGN=MIDDLE  ALT=\"500%2Ae%5E%280.08%2A5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >= $<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=500%2Ae%5E0.4 ALIGN=MIDDLE  ALT=\"500%2Ae%5E0.4\"   DISABLED_style= \"max-width:90%; height:auto;\" >= $<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=745.91 ALIGN=MIDDLE  ALT=\"745.91\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );