document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=117059>Question 117059</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose that 2000.00 is invested at a rate of 6% per year compounded continuously. What is the balance after 1 year?  After 2 years </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #85144 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=85144\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The formula for continuous compounding is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+C%2Ae%5E%28r%2At%29 ALIGN=MIDDLE  ALT=\"P+=+C%2Ae%5E%28r%2At%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "in which P is the total future value, C is the initial deposit of money, e is the base of the<BR>\n" );
document.write( "natural logarithms and is approximately 2.718281828, r is the annual interest rate expressed <BR>\n" );
document.write( "as a decimal, and t is the number of years of the investment.<BR>\n" );
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document.write( "For this problem you are told that the initial deposit of money is $2000. Substituting<BR>\n" );
document.write( "this value for C in the equation results in the equation becoming:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+2000%2Ae%5E%28r%2At%29 ALIGN=MIDDLE  ALT=\"P+=+2000%2Ae%5E%28r%2At%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You are also told that the annual interest rate is 6%. In decimal form this is 0.06. Substitute<BR>\n" );
document.write( "this decimal for r in the equation to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+2000%2Ae%5E%280.06%2At%29 ALIGN=MIDDLE  ALT=\"P+=+2000%2Ae%5E%280.06%2At%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now that we are this far, we can substitute the two values of t that you are asked to<BR>\n" );
document.write( "find the future value (P) for. In the first case you are asked to find the future value<BR>\n" );
document.write( "of the $2000 initial deposit if it is left in the investment for 1 year. Substitute 1 for<BR>\n" );
document.write( "t in the equation and the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+2000%2Ae%5E%280.06%2A1%29 ALIGN=MIDDLE  ALT=\"P+=+2000%2Ae%5E%280.06%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now it's basically a calculator problem. The exponent is 0.06 times 1 so it just equals 0.06<BR>\n" );
document.write( "and the problem is then:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+2000%2Ae%5E%280.06%29 ALIGN=MIDDLE  ALT=\"P+=+2000%2Ae%5E%280.06%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "If you use your calculator to raise e (or 2.718281828) to the exponent 0.06 you find that <BR>\n" );
document.write( "it is 1.061836547 and the equation reduces to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+2000+%2A+1.061836547+=+2123.673093 ALIGN=MIDDLE  ALT=\"P+=+2000+%2A+1.061836547+=+2123.673093\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "When rounded off the answer is that using continuous compounding at an annual rate of 6%<BR>\n" );
document.write( "a $2000 initial investment will be worth $2123.67 at the end of one year.<BR>\n" );
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document.write( "If the $2000 is left on deposit for 2 years, the only thing that changes in the equation is the <BR>\n" );
document.write( "value of t. For this calculation, t = 2 and the equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+2000%2Ae%5E%280.06%2At%29 ALIGN=MIDDLE  ALT=\"P+=+2000%2Ae%5E%280.06%2At%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+2000%2Ae%5E%280.06%2A2%29+=+2000%2Ae%5E%280.12%29 ALIGN=MIDDLE  ALT=\"P+=+2000%2Ae%5E%280.06%2A2%29+=+2000%2Ae%5E%280.12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Using your calculator on <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E0.12 ALIGN=MIDDLE  ALT=\"e%5E0.12\"   DISABLED_style= \"max-width:90%; height:auto;\" > you should find that it equals 1.127496852 and this reduces<BR>\n" );
document.write( "the equation to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+2000%2A1.127496852+=+2254.993703 ALIGN=MIDDLE  ALT=\"P+=+2000%2A1.127496852+=+2254.993703\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "When rounded off the answer is that using continuous compounding at an annual rate of 6%<BR>\n" );
document.write( "a $2000 initial investment will be worth $2254.99 at the end of two years.<BR>\n" );
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document.write( "Hope this helps you to understand how to do continuous compounding for an investment.<BR>\n" );
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