document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1189542>Question 1189542</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An initial population of 30 fish is introduced into a lake. This fish population grows according to a continuous exponential growth model. There are 69 fish in the lake after 12 years.<BR>\n" );
document.write( "(a) Let “t” be the time (in years) since the initial population is introduced, and let y be the number of fish at time t. Write a formula relating y to t. <BR>\n" );
document.write( "Use exact expressions to fill in the missing parts of the formula. Do not use approximations.\r<BR>\n" );
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document.write( "(b) How many fish are there 14 years after the initial population is<BR>\n" );
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document.write( "Do not round any intermediate computations, and round your<BR>\n" );
document.write( "answer to the nearest whole number. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #820918 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=820918\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the continuous exponential growth formula is:<BR>\n" );
document.write( "f = p * e ^ (r * t)<BR>\n" );
document.write( "f is the future value.<BR>\n" );
document.write( "p is the present value.<BR>\n" );
document.write( "e is the scientific constant e which is equal to 2.718281828.....<BR>\n" );
document.write( "r is the interest rate per time period.<BR>\n" );
document.write( "t is the number of time periods.\r<BR>\n" );
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document.write( "for this problem, the time periods are in years.\r<BR>\n" );
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document.write( "the initial population is 30.<BR>\n" );
document.write( "12 years later, the population is 69.<BR>\n" );
document.write( "f = 69<BR>\n" );
document.write( "p = 30<BR>\n" );
document.write( "t = 12<BR>\n" );
document.write( "r equals what you want to find.\r<BR>\n" );
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document.write( "the formula becomes:\r<BR>\n" );
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document.write( "69 = 30 * e ^ (r * 12)\r<BR>\n" );
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document.write( "divide both sides of the equation by 30 to get:<BR>\n" );
document.write( "69/30 = e ^ (r * 12<BR>\n" );
document.write( "take the natural log of both sides of the equation to get:<BR>\n" );
document.write( "ln(69/30) = ln(e ^ (r * 12)<BR>\n" );
document.write( "since ln(e ^ x) = x * ln(e) and since ln(e) = 1, this becomes:<BR>\n" );
document.write( "ln(69/30) = 12 * r<BR>\n" );
document.write( "divive both sides of this equation by 12 to get:<BR>\n" );
document.write( "ln(69/30) / 12 = r<BR>\n" );
document.write( "solve for r to get:<BR>\n" );
document.write( "r = .0694090936.\r<BR>\n" );
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document.write( "confirm by replacing r with that in the original equationand solving for f to get:<BR>\n" );
document.write( "f = 30 * e ^ (.0694090936 * 12) = 69.\r<BR>\n" );
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document.write( "this confirms the value of r is correct.\r<BR>\n" );
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document.write( "to determine how many fish are there after 14 years, the equation becomes:<BR>\n" );
document.write( "f = 30 * e ^ (.0694090936 * 14) = 79.27514835.\r<BR>\n" );
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document.write( "this equation can be graphed.<BR>\n" );
document.write( "it looks like this:\r<BR>\n" );
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document.write( " </SPAN>\n" );
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