document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1187605>Question 1187605</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A mutual fund has a growth rate recorded each year, expressed by the function  <BR>\n" );
document.write( "𝑉′(𝑡)=𝑟(𝑡)=200𝑒0.04𝑡, 0≤𝑡≤10 (in thousands of dollars per year), where 𝑡 is the number of years passed from year 1997. The value of the fund in year 1997 is $1,000,000. \r<BR>\n" );
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document.write( "a) Determine the function 𝑉(𝑡) which represents the value of the fund 𝑡 years after year 1997. <BR>\n" );
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document.write( "b) Determine the value of the fund in year 2007.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849310 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849310\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a) Determining the function V(t):**\r<BR>\n" );
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document.write( "To find the function V(t) that represents the value of the fund *t* years after 1997, we need to integrate the growth rate function V'(t) = r(t).\r<BR>\n" );
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document.write( "∫V'(t) dt = ∫200e^(0.04t) dt\r<BR>\n" );
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document.write( "V(t) = (200/0.04)e^(0.04t) + C\r<BR>\n" );
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document.write( "V(t) = 5000e^(0.04t) + C\r<BR>\n" );
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document.write( "We know that in 1997 (t=0), the value of the fund is $1,000,000.  We can use this information to find the constant of integration, C.\r<BR>\n" );
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document.write( "1,000,000 = 5000e^(0.04*0) + C\r<BR>\n" );
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document.write( "1,000,000 = 5000 + C\r<BR>\n" );
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document.write( "C = 995,000\r<BR>\n" );
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document.write( "Therefore, the function representing the value of the fund *t* years after 1997 is:\r<BR>\n" );
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document.write( "V(t) = 5000e^(0.04t) + 995,000\r<BR>\n" );
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document.write( "**b) Determining the value of the fund in 2007:**\r<BR>\n" );
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document.write( "The year 2007 is 10 years after 1997, so we need to find V(10):\r<BR>\n" );
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document.write( "V(10) = 5000e^(0.04*10) + 995,000\r<BR>\n" );
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document.write( "V(10) = 5000e^(0.4) + 995,000\r<BR>\n" );
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document.write( "V(10) ≈ 5000 * 1.4918 + 995,000\r<BR>\n" );
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document.write( "V(10) ≈ 7459 + 995,000\r<BR>\n" );
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document.write( "V(10) ≈ 1,002,459\r<BR>\n" );
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document.write( "Therefore, the value of the fund in 2007 is approximately $1,002,459.<BR>\n" );
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