document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=929310>Question 929310</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1.  You have deposited $7,000 in an account that pays 6.25% interest compounded continuously.  How long will it take your money to double?\r<BR>\n" );
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document.write( "2.  You have deposited $12,000 in an account compounded monthly.  After 7 years the balance has grown to $15,532.  Find the rate.\r<BR>\n" );
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document.write( "PLEASE HELP ME ANSWER THESE 2 QUESTIONS!!!  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #564307 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=564307\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1. For continuous compounding, the balance amount <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > is related to<BR>\n" );
document.write( "the principal (initial investment <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P ALIGN=MIDDLE  ALT=\"P\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "the rate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > (written as a decimal), and<BR>\n" );
document.write( "the time in years, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > , by the formula<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=P%2Ae%5E%28%28r%2At%29%29 ALIGN=MIDDLE  ALT=\"A=P%2Ae%5E%28%28r%2At%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The number <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;\" > that come up pretty often . It is not invented to make students learn one more number; it just comes up naturally.<BR>\n" );
document.write( "We use it as a base for powers, and of course for logarithms that we call natural logarithms <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6.25%25=6.25%2F100=0.0625 ALIGN=MIDDLE  ALT=\"6.25%25=6.25%2F100=0.0625\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So, for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P=7000 ALIGN=MIDDLE  ALT=\"P=7000\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=0.0625 ALIGN=MIDDLE  ALT=\"r=0.0625\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=2%2AP=2%2A7000 ALIGN=MIDDLE  ALT=\"A=2%2AP=2%2A7000\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we can write<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2AP=P%2Ae%5E0.0625t ALIGN=MIDDLE  ALT=\"2%2AP=P%2Ae%5E0.0625t\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2=e%5E0.0625t ALIGN=MIDDLE  ALT=\"2=e%5E0.0625t\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Note that we did not need to know the amount invested.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2=e%5E0.0625t ALIGN=MIDDLE  ALT=\"2=e%5E0.0625t\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%282%29=0.0625t ALIGN=MIDDLE  ALT=\"ln%282%29=0.0625t\"   DISABLED_style= \"max-width:90%; height:auto;\" > (taking natural logarithms on both sides).<BR>\n" );
document.write( "Using the approximate value <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%282%29=0.693147 ALIGN=MIDDLE  ALT=\"ln%282%29=0.693147\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.0625t=0.693147 ALIGN=MIDDLE  ALT=\"0.0625t=0.693147\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0.693147%2F.0625 ALIGN=MIDDLE  ALT=\"t=0.693147%2F.0625\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=11.09 ALIGN=MIDDLE  ALT=\"t=11.09\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded).<BR>\n" );
document.write( "So, it will take 11 years for the money to double.\r<BR>\n" );
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document.write( "2. $12,000 compounded monthly. After 7 years the balance has grown to $15,532. Find the rate.<BR>\n" );
document.write( "For monthly compounding, the balance grow by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F12%29 ALIGN=MIDDLE  ALT=\"%281%2F12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the yearly rate every month.<BR>\n" );
document.write( "So over <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7years=7%2A12months=84months ALIGN=MIDDLE  ALT=\"7years=7%2A12months=84months\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the balance grew by a factor of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B%28rate%2F12%29 ALIGN=MIDDLE  ALT=\"1%2B%28rate%2F12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > applied <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=84 ALIGN=MIDDLE  ALT=\"84\"   DISABLED_style= \"max-width:90%; height:auto;\" > times to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15320=12000%2A%281%2Brate%2F12%29%5E84 ALIGN=MIDDLE  ALT=\"15320=12000%2A%281%2Brate%2F12%29%5E84\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Solving for rate:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15320=12000%2A%281%2Brate%2F12%29%5E84 ALIGN=MIDDLE  ALT=\"15320=12000%2A%281%2Brate%2F12%29%5E84\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15320%2F12000=%281%2Brate%2F12%29%5E84 ALIGN=MIDDLE  ALT=\"15320%2F12000=%281%2Brate%2F12%29%5E84\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.2766666667=%281%2Brate%2F12%29%5E84 ALIGN=MIDDLE  ALT=\"1.2766666667=%281%2Brate%2F12%29%5E84\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%281.2766666667%29=84%2Alog%281%2Brate%2F12%29 ALIGN=MIDDLE  ALT=\"log%281.2766666667%29=84%2Alog%281%2Brate%2F12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F84%29%2Alog%281.2766666667%29=log%281%2Brate%2F12%29 ALIGN=MIDDLE  ALT=\"%281%2F84%29%2Alog%281.2766666667%29=log%281%2Brate%2F12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using logarithm base 10 for an approximate calculation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F84%29%2A0.10677519=log%281%2Brate%2F12%29 ALIGN=MIDDLE  ALT=\"%281%2F84%29%2A0.10677519=log%281%2Brate%2F12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.0012628=log%281%2Brate%2F12%29 ALIGN=MIDDLE  ALT=\"0.0012628=log%281%2Brate%2F12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Reversing the logarithm by calculating power of 10 on both sides of the equal sign:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%5E0.0012628=1%2Brate%2F12 ALIGN=MIDDLE  ALT=\"10%5E0.0012628=1%2Brate%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.002912=1%2Brate%2F12 ALIGN=MIDDLE  ALT=\"1.002912=1%2Brate%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.002912-1=rate%2F12 ALIGN=MIDDLE  ALT=\"1.002912-1=rate%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.002912=rate%2F12 ALIGN=MIDDLE  ALT=\"0.002912=rate%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=rate=0.03496=%223.494%25%22 ALIGN=MIDDLE  ALT=\"rate=0.03496=%223.494%25%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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