document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=27770>Question 27770</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The perimeter of a rectangle is 64cm. The area is 240cm square. What are its dimension? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #15147 by </B><A HREF=/tutors/aboutme.mpl?userid=sdmmadam@yahoo.com><B>sdmmadam@yahoo.com(530)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=sdmmadam@yahoo.com><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=15147\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The perimeter of a rectangle is 64cm. The area is 240cm square. <BR>\n" );
document.write( "What are its dimensions?<BR>\n" );
document.write( "Let the dimensions of the rectangle be length= a cms and width = b cms<BR>\n" );
document.write( "Formula for area of a rectangle is <BR>\n" );
document.write( "Area = lenght X width = (a)X(b) = (ab) sqcms<BR>\n" );
document.write( "Formula for Perimeter of a rectangle is <BR>\n" );
document.write( "Perimeter = 2(length + width)= 2(a+b)cms<BR>\n" );
document.write( "By data 2(a+b)= 64 which means (a+b) = 32 ----(1) (on division by 2)<BR>\n" );
document.write( "and  (ab) = 240  ----(2)<BR>\n" );
document.write( "We have by formula  (a-b)^2 = (a+b)^2-4(ab)  ----(*)<BR>\n" );
document.write( "Therefore using (1) and (2) in (*)<BR>\n" );
document.write( "(a-b)^2 = (a+b)^2-4(ab)  = (32)^2 - 4X(240) = 1024 - 960 = 64<BR>\n" );
document.write( "(a-b)^2 = 64<BR>\n" );
document.write( "which implies (a-b) = 8 ----(3) (taking the positive root)<BR>\n" );
document.write( "Now consider (a+b) = 32 ----(1) <BR>\n" );
document.write( "Adding (3) and (1)<BR>\n" );
document.write( "[(a-b) + (a+b) ] = (8 + 32)<BR>\n" );
document.write( " a-b+a+b = 40<BR>\n" );
document.write( " a+a-b+b = 40  (by additive associativity)<BR>\n" );
document.write( " 2a + 0 =40<BR>\n" );
document.write( " 2a = 40<BR>\n" );
document.write( "a = 40/2 = 20<BR>\n" );
document.write( "Putting a= 20 in (1)<BR>\n" );
document.write( "a+b = 32<BR>\n" );
document.write( "20 + b = 32 which means b = (32-20) = 12<BR>\n" );
document.write( "Therefore length= 20 cms and width = 12 cms<BR>\n" );
document.write( "Verification: Area = length X width =  20X12 = 240  which is correct<BR>\n" );
document.write( "And Perimeter = 2(lenght + width) = 2X(20 +12) = 2X32 = 64 which is correct.<BR>\n" );
document.write( "Note: In the above, from (a-b)^2 = 64<BR>\n" );
document.write( "we have taken the positive root 8 for (a-b).<BR>\n" );
document.write( "What if we consider the negative root (-8) because (-8)^2 also = 64<BR>\n" );
document.write( "Okay, let us consider  (a-b) = -8 say ----(4)<BR>\n" );
document.write( "Now adding (4) and (1), 2a = 32-8 = 24 which means a = 12 which implies b = 20<BR>\n" );
document.write( "So we get the dimensions the same but only the tags are different.<BR>\n" );
document.write( "So in such problems it it enough if we consider the positive root.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );