document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=211874>Question 211874</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0201419513/algebrahomeworkh>just-in-time algebra and trigonometry</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> a 2 ft. wire is cut into two pieces. one of the pieces, of length x, is bent into a circle. the other piece is bent into a rectangle whose length is twice the size of its width. what is the total area of the two shapes as a function of x. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #160080 by </B><A HREF=/tutors/aboutme.mpl?userid=nerdybill><B>nerdybill(7384)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=nerdybill><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=nerdybill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=160080\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">     a 2 ft. wire is cut into two pieces. one of the pieces, of length x, is bent into a circle. the other piece is bent into a rectangle whose length is twice the size of its width. what is the total area of the two shapes as a function of x.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Let x = circumference of circle<BR>\n" );
document.write( "then<BR>\n" );
document.write( "2-x = perimeter of rectangle<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Since for any circle:<BR>\n" );
document.write( "circumference = (pi)d = 2(pi)r<BR>\n" );
document.write( "then<BR>\n" );
document.write( "x = 2(pi)r<BR>\n" );
document.write( "r = x/(2(pi))<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Area of circle:<BR>\n" );
document.write( "(pi)r^2<BR>\n" );
document.write( "=(pi)[x/(2(pi))]^2]<BR>\n" );
document.write( "=(pi)[x^2/(4(pi)^2)]<BR>\n" );
document.write( "=x^2/(4(pi))<BR>\n" );
document.write( ".<BR>\n" );
document.write( "rectangle:<BR>\n" );
document.write( "Let w = width<BR>\n" );
document.write( "then<BR>\n" );
document.write( "2w = length<BR>\n" );
document.write( "perimeter = 2(w + 2w)<BR>\n" );
document.write( "perimeter = 2(3w)<BR>\n" );
document.write( "2-x = 2(3w)<BR>\n" );
document.write( "2-x = 6w<BR>\n" );
document.write( "(2-x)/6 = w (width)<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Length:<BR>\n" );
document.write( "2w = 2[(2-x)/6] = (2-x)/3 (length)<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Area of rectangle:<BR>\n" );
document.write( "w*L<BR>\n" );
document.write( "= (2-x)/6 * (2-x)/3<BR>\n" );
document.write( "= (2-x)^2/18<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Total area:<BR>\n" );
document.write( "\"area of circle\" + \"area of rectangle\"<BR>\n" );
document.write( "= x^2/(4(pi)) + (2-x)^2/18<BR>\n" );
document.write( "= 18x^2/(72(pi)) + (4(pi))(2-x)^2/(72(pi))<BR>\n" );
document.write( "= [18x^2 + (4(pi))(2-x)^2]/(72(pi))<BR>\n" );
document.write( "= [18x^2 + (4(pi))(2-x)(2-x)]/(72(pi))<BR>\n" );
document.write( "= [18x^2 + (4(pi))(4-4x+x^2)]/(72(pi))<BR>\n" );
document.write( "= [18x^2 + 16(pi) - 16(pi)x + 4(pi)x^2]/(72(pi))<BR>\n" );
document.write( "= [18x^2 + 4(pi)x^2 + 16(pi) - 16(pi)x]/(72(pi))<BR>\n" );
document.write( "= [2x^2(9 + 2(pi)) + 16(pi)(1-x)]/(72(pi))<BR>\n" );
document.write( "= 2[x^2(9 + 2(pi)) + 8(pi)(1-x)]/(72(pi))<BR>\n" );
document.write( "= [x^2(9 + 2(pi)) + 8(pi)(1-x)]/(36(pi))<BR>\n" );
document.write( "= [x^2(9 + 2(pi)) + 8(pi)(1-x)]/(36(pi))\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );