document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171575>Question 1171575</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A botanist starts with 3 plants. She takes 5 cuttings from each plant<BR>\n" );
document.write( "to start new plants. Later, she takes 5 cuttings from each new plant, and so on.<BR>\n" );
document.write( "a) Draw a graph of the number of new plants in each of the first 5 rounds of cuttings.<BR>\n" );
document.write( "b) Write an equation to model the number of new plants, P, in the nth round of cuttings.<BR>\n" );
document.write( "c) How would the graph and equation change in each scenario? Explain.<BR>\n" );
document.write( "i) The botanist starts with 10 plants.<BR>\n" );
document.write( "ii) In each round, the botanist takes 2 cuttings from each plant. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850894 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=850894\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break this down step by step:\r<BR>\n" );
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document.write( "**a) Graph of New Plants**\r<BR>\n" );
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document.write( "* **Round 1:**<BR>\n" );
document.write( "    * Starts with 3 plants.<BR>\n" );
document.write( "    * Takes 5 cuttings from each: 3 plants * 5 cuttings/plant = 15 new plants.<BR>\n" );
document.write( "* **Round 2:**<BR>\n" );
document.write( "    * Starts with 15 plants.<BR>\n" );
document.write( "    * Takes 5 cuttings from each: 15 plants * 5 cuttings/plant = 75 new plants.<BR>\n" );
document.write( "* **Round 3:**<BR>\n" );
document.write( "    * Starts with 75 plants.<BR>\n" );
document.write( "    * Takes 5 cuttings from each: 75 plants * 5 cuttings/plant = 375 new plants.<BR>\n" );
document.write( "* **Round 4:**<BR>\n" );
document.write( "    * Starts with 375 plants.<BR>\n" );
document.write( "    * Takes 5 cuttings from each: 375 * 5 = 1875<BR>\n" );
document.write( "* **Round 5:**<BR>\n" );
document.write( "    * Starts with 1875 plants.<BR>\n" );
document.write( "    * Takes 5 cuttings from each: 1875 * 5 = 9375.\r<BR>\n" );
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document.write( "Graph:\r<BR>\n" );
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document.write( "* X-axis: Round number (1, 2, 3, 4, 5)<BR>\n" );
document.write( "* Y-axis: Number of new plants (15, 75, 375, 1875, 9375)\r<BR>\n" );
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document.write( "The graph will show an exponential growth curve.\r<BR>\n" );
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document.write( "**b) Equation to Model New Plants (P)**\r<BR>\n" );
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document.write( "* We see a pattern:<BR>\n" );
document.write( "    * Round 1: 3 * 5^1 = 15<BR>\n" );
document.write( "    * Round 2: 3 * 5^2 = 75<BR>\n" );
document.write( "    * Round 3: 3 * 5^3 = 375<BR>\n" );
document.write( "    * Round 4: 3*5^4 = 1875<BR>\n" );
document.write( "    * Round 5: 3*5^5 = 9375<BR>\n" );
document.write( "* Therefore, the equation is: P = 3 * 5^n, where P is the number of new plants and n is the round number.\r<BR>\n" );
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document.write( "**c) Changes in Graph and Equation**\r<BR>\n" );
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document.write( "**i) Starting with 10 Plants:**\r<BR>\n" );
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document.write( "* **Equation:**<BR>\n" );
document.write( "    * The initial factor changes from 3 to 10.<BR>\n" );
document.write( "    * P = 10 * 5^n<BR>\n" );
document.write( "* **Graph:**<BR>\n" );
document.write( "    * The y-intercept (starting point) of the graph will be higher.<BR>\n" );
document.write( "    * The exponential growth will be steeper, as there are more plants at each round.\r<BR>\n" );
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document.write( "**ii) Taking 2 Cuttings from Each Plant:**\r<BR>\n" );
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document.write( "* **Equation:**<BR>\n" );
document.write( "    * The base of the exponent changes from 5 to 2.<BR>\n" );
document.write( "    * P = 3 * 2^n<BR>\n" );
document.write( "* **Graph:**<BR>\n" );
document.write( "    * The graph will still show exponential growth, but it will be less steep.<BR>\n" );
document.write( "    * The number of new plants will increase at a slower rate compared to 5 cuttings per plant.<BR>\n" );
document.write( " </SPAN>\n" );
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