document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1168648>Question 1168648</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A person invests $4050 in stocks today, and their value doubles every 4 years. Let f (t) be the value (in dollars) of the investment at t years from now <BR>\n" );
document.write( "A. Find an equation of f.<BR>\n" );
document.write( "B Find the value of the investment 20 years from now<BR>\n" );
document.write( "a. f can be modeled by the exponential equation of the form f(t)=ab^t with the following values kof a and b.<BR>\n" );
document.write( "THE ANSWERS I NEED TO FIND ARE: <BR>\n" );
document.write( "a=<BR>\n" );
document.write( "b= </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #793221 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=793221\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you are given that the investment doubles every 4 years.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "since your investment period is 20 years, then 4 goes into 20 five times.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your future value will be 4050*2^5 = 129,600.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "if you want to model this using continuous compounding formula, you would do the following.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the continuous compounding formula is f = p*e^(r*t)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "f is the future value and p is the present value and r is the interest rate per time period and t is the number of time periods.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "to use this formula, consider p = 4050 and f = 129600 amd t = 20 years.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the formula becomes 129600 = 4050*e^(r*20)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "divide both sides of this formula by 4050 to get:<BR>\n" );
document.write( "129600/4050 = e^(r*20)<BR>\n" );
document.write( "take the natural log of both sides of this formula to get:<BR>\n" );
document.write( "ln(129600/4050) = ln(e^(r*20))<BR>\n" );
document.write( "since ln(e^(r*20)) is equal to r*20*ln(e), the formula becomes:<BR>\n" );
document.write( "ln(129600/4050) = r*20*ln(e)<BR>\n" );
document.write( "since ln(e) = 1, the formula becomes:<BR>\n" );
document.write( "ln(129600/4050) = r*20<BR>\n" );
document.write( "divide both sides of this formula by 20 to get:<BR>\n" );
document.write( "ln(129600/4050)/20 = r<BR>\n" );
document.write( "solve for r to get:<BR>\n" );
document.write( "r = .1732867951.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "to confirm, replace r in the formula with that to get:<BR>\n" );
document.write( "129600 = 4050 * e ^ (.1732867951*20) becomes 129600 = 129600.<BR>\n" );
document.write( "this confirms the value of r is correct.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "with discrete compounding, the formula used is f = p*(1+r)^n\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "f is the future value, p is the present value, r is the interest rate per time period, n is the number of time periods.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "assuming the investment doubles every 4 years, and assuming that the money is compounding annually, you would do the following:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when the money doubles every 4 years, f = 2, p = 1, r = what you want to find, n = 4\r<BR>\n" );
document.write( "\n" );
document.write( "the formula becomes 2 = 1*(1+r)^4<BR>\n" );
document.write( "simplify to get 2 = (1+r)^4<BR>\n" );
document.write( "take the 4th root of both sides of this equation to get:<BR>\n" );
document.write( "(2)^(1/4)=1+r<BR>\n" );
document.write( "subtract 1 from both sides of this equation to get:<BR>\n" );
document.write( "(2)^(1/4)-1 = r<BR>\n" );
document.write( "solve for r to get:<BR>\n" );
document.write( "r = .189207115.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that's the interest rate per year.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the discrete compounding formula becomes f = 4050*(1+.189207115)^20\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "f is the future value<BR>\n" );
document.write( "p is the present value<BR>\n" );
document.write( "r = .189207115 per year.<BR>\n" );
document.write( "n = 20 years.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "solve for f to get f = 129600.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you can graph all three of these equations to get the same answer.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "here's the graph.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2020/103004.jpg\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the red graph has each interval of x representing 4 years.<BR>\n" );
document.write( "therefore, 20 years is when x = 5.<BR>\n" );
document.write( "the blue graph has each interval of x representing 1 year.<BR>\n" );
document.write( "therefore, 20 years is when x = 20.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the blue graph represents both the discrete compounding formula and the continuous compounding formula.<BR>\n" );
document.write( "since these two formulas are equivalent to each other, the same blue graph represents both.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "any questions about any of this, let me know and i'll answer as best and as soon as i can.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "theo\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );