document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198079>Question 1198079</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A rectangle has an area of 300 square centimetres. suppose the width increases from 25 to 30 cm, but the area stays the same. <BR>\n" );
document.write( "a) determine the fractional change in the width.<BR>\n" );
document.write( "b) determine the fractional change in the length (r) using the constant product principle. <BR>\n" );
document.write( "c) Determine the fractional change in the length (r) if the width increases by the fraction, w.<BR>\n" );
document.write( "d) Sketch the graph that represents the fractional change in the length as a function of the fractional change in the width.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848348 by </B><A HREF=/tutors/aboutme.mpl?userid=onyulee><B>onyulee(41)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=onyulee><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848348\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a) Determine the fractional change in the width.**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Original Width:** 25 cm<BR>\n" );
document.write( "* **New Width:** 30 cm<BR>\n" );
document.write( "* **Change in Width:** 30 cm - 25 cm = 5 cm<BR>\n" );
document.write( "* **Fractional Change in Width:** (Change in Width) / (Original Width) = 5 cm / 25 cm = 1/5 or 0.20\r<BR>\n" );
document.write( "\n" );
document.write( "**b) Determine the fractional change in the length (r) using the constant product principle.**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Constant Product Principle:** For a constant area, if one dimension increases, the other dimension must decrease proportionally.\r<BR>\n" );
document.write( "\n" );
document.write( "* **Original Area:** 300 cm²<BR>\n" );
document.write( "* **Original Width:** 25 cm<BR>\n" );
document.write( "* **Original Length:** 300 cm² / 25 cm = 12 cm\r<BR>\n" );
document.write( "\n" );
document.write( "* **New Width:** 30 cm<BR>\n" );
document.write( "* **New Length:** 300 cm² / 30 cm = 10 cm\r<BR>\n" );
document.write( "\n" );
document.write( "* **Change in Length:** 10 cm - 12 cm = -2 cm<BR>\n" );
document.write( "* **Fractional Change in Length (r):** (-2 cm) / 12 cm = -1/6 or -0.1667\r<BR>\n" );
document.write( "\n" );
document.write( "**c) Determine the fractional change in the length (r) if the width increases by the fraction, w.**\r<BR>\n" );
document.write( "\n" );
document.write( "* Let 'w' be the fractional increase in width.<BR>\n" );
document.write( "* New Width = Original Width * (1 + w)\r<BR>\n" );
document.write( "\n" );
document.write( "* Since Area = Width * Length:<BR>\n" );
document.write( "    * Original Width * Original Length = New Width * New Length<BR>\n" );
document.write( "    * Original Width * Original Length = (Original Width * (1 + w)) * New Length<BR>\n" );
document.write( "    * New Length = (Original Length) / (1 + w)\r<BR>\n" );
document.write( "\n" );
document.write( "* Fractional Change in Length (r) = (New Length - Original Length) / Original Length<BR>\n" );
document.write( "    * r = [(Original Length) / (1 + w) - Original Length] / Original Length<BR>\n" );
document.write( "    * r = [1 / (1 + w) - 1] <BR>\n" );
document.write( "    * r = [1 - (1 + w)] / (1 + w)<BR>\n" );
document.write( "    * r = -w / (1 + w)\r<BR>\n" );
document.write( "\n" );
document.write( "**d) Sketch the graph that represents the fractional change in the length as a function of the fractional change in the width.**\r<BR>\n" );
document.write( "\n" );
document.write( "* This equation (r = -w / (1 + w)) represents a hyperbola. <BR>\n" );
document.write( "* Key features of the graph:<BR>\n" );
document.write( "    * As 'w' (fractional change in width) increases, 'r' (fractional change in length) becomes increasingly negative.<BR>\n" );
document.write( "    * When 'w' is 0 (no change in width), 'r' is also 0.<BR>\n" );
document.write( "    * The graph will approach an asymptote at 'w' = -1, as the length would become infinitely large.\r<BR>\n" );
document.write( "\n" );
document.write( "**In Summary:**\r<BR>\n" );
document.write( "\n" );
document.write( "* The fractional change in the width is 0.20.<BR>\n" );
document.write( "* The fractional change in the length is -0.1667.<BR>\n" );
document.write( "* The general relationship between the fractional change in width (w) and the fractional change in length (r) is given by: r = -w / (1 + w).<BR>\n" );
document.write( "* The graph of this relationship is a hyperbola.\r<BR>\n" );
document.write( "\n" );
document.write( "I hope this comprehensive explanation is helpful!<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );