document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=251449>Question 251449</A>: <I><SPAN STYLE=\"word-break: break-all;\"> P&#8320; is invested in a savings account where interest is compounded continuously at 3.1% per year.<BR>\n" );
document.write( "a)	Express P(t) in terms of P&#8320; and 0.031<BR>\n" );
document.write( "b)	Suppose that 1000$ is invested, what is the balance after 1 yr? 2 yrs?<BR>\n" );
document.write( "c)	When will an investment of 1000$ double itself?\r<BR>\n" );
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document.write( "Helpful formula:<BR>\n" );
document.write( "When an amount of money P&#8320; is invested at interest rate k, compounded continuously, interest is computed every instant and added to the original amount. The balance P(t), after t years, is given by the exponential growth model\r<BR>\n" );
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document.write( "				P(t)=P&#8320;e^kt<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 #183341 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=183341\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the formula for continuous compounding is:\r<BR>\n" );
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document.write( "f = p*e^(r*t)\r<BR>\n" );
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document.write( "f = future value<BR>\n" );
document.write( "p = present amount<BR>\n" );
document.write( "e = scientific constant whose value is equal to 2.718281828...<BR>\n" );
document.write( "r = interest rate per year<BR>\n" );
document.write( "t = number of years\r<BR>\n" );
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document.write( "note:\r<BR>\n" );
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document.write( "interest rate percent / 100% = interest rate.\r<BR>\n" );
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document.write( "in your problem, the interest rate per year would be:\r<BR>\n" );
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document.write( "3.1% / 100% = .031\r<BR>\n" );
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document.write( "p = $1,000.00\r<BR>\n" );
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document.write( "for a value after 1 year, t = 1<BR>\n" );
document.write( "for a value after 2 years, t = 2<BR>\n" );
document.write( "how to solve for when the investment doubles comes after that.\r<BR>\n" );
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document.write( "value of $1,000 investment after 1 year.\r<BR>\n" );
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document.write( "p = $1,000<BR>\n" );
document.write( "r = .031<BR>\n" );
document.write( "t = 1\r<BR>\n" );
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document.write( "f = p * e^(r*t) becomes:\r<BR>\n" );
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document.write( "f = 1000 * e^(.031*1)\r<BR>\n" );
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document.write( "solve for f to get:\r<BR>\n" );
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document.write( "f = 1031.485504\r<BR>\n" );
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document.write( "value of $1,000 investment after 2 years.\r<BR>\n" );
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document.write( "p = $1,000<BR>\n" );
document.write( "r = .031<BR>\n" );
document.write( "t = 2\r<BR>\n" );
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document.write( "f = p * e^(r*t) becomes:\r<BR>\n" );
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document.write( "f = 1000 * e^(.031*2)\r<BR>\n" );
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document.write( "solve for f to get:\r<BR>\n" );
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document.write( "f = 1063.962345\r<BR>\n" );
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document.write( "how long does it take for the money to double.\r<BR>\n" );
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document.write( "let p = 1<BR>\n" );
document.write( "let f = 2\r<BR>\n" );
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document.write( "formula of f = p * e^(r*t) becomes:\r<BR>\n" );
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document.write( "2 = 1 * e^(.031*t)\r<BR>\n" );
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document.write( "you need to solve for t.\r<BR>\n" );
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document.write( "2 = 1 * e^(.031*t) becomes:\r<BR>\n" );
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document.write( "2 = e^(.031*t)\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( "log(2) = log(e^(.031*t))\r<BR>\n" );
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document.write( "since log (a^b) = b*log(a), your equation becomes:\r<BR>\n" );
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document.write( "log(2) = .031*t*log(e)\r<BR>\n" );
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document.write( "divide both sides of this equation by .031 * log(3) to get\r<BR>\n" );
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document.write( "t = log(2) / (.031*log(e))\r<BR>\n" );
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document.write( "solve for t to get:\r<BR>\n" );
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document.write( "t = 22.35958647\r<BR>\n" );
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document.write( "confirm by substituting for t in the original equation.\r<BR>\n" );
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document.write( "2 = 1 * e^(.031*t) becomes:\r<BR>\n" );
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document.write( "2 = 1 * e^(.031*22.35958647) which becomes:\r<BR>\n" );
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document.write( "2 = 1 * e^(.693147181) which becomes:\r<BR>\n" );
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document.write( "2 = 1 * 2 which becomes:\r<BR>\n" );
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document.write( "2 = 2 confirming that the solution is valid.\r<BR>\n" );
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document.write( "your answer is:\r<BR>\n" );
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document.write( "the investment will double in 22.35958647 years.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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