document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=313288>Question 313288</A>: <I><SPAN STYLE=\"word-break: break-all;\">  Write an equation and solve.  Using guess & check method only.  \r<BR>\n" );
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document.write( "The width of Henry's pool is 8 yards less than its length.  If the area of the pool is 200 square yards, what are its length and width?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #224062 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=224062\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The formula for the area of a rectangle is L * W = A\r<BR>\n" );
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document.write( "Since the width of the pool is 8 yards less than its length, we get:\r<BR>\n" );
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document.write( "L = Length of Pool<BR>\n" );
document.write( "W = L - 8 = Width of Pool\r<BR>\n" );
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document.write( "We know the area of the pool is 200 square yards.\r<BR>\n" );
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document.write( "This means that L * W = 200\r<BR>\n" );
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document.write( "Since we also know that W = L - 8, then this formula becomes:\r<BR>\n" );
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document.write( "L * (L - 8) = 200\r<BR>\n" );
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document.write( "We simplify this formula to get:\r<BR>\n" );
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document.write( "L^2 - 8*L = 200\r<BR>\n" );
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document.write( "We subtract 200 from both sides of this equation to get:\r<BR>\n" );
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document.write( "L^2 - 8*L - 200 = 0\r<BR>\n" );
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document.write( "From here we could solve the quadratic equation or we could do guess and check.\r<BR>\n" );
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document.write( "In order to do guess and check, we'll go back to:\r<BR>\n" );
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document.write( "L^2 - 8*L = 200 since solving for that is one step less.\r<BR>\n" );
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document.write( "Allow L = 10 to start with.\r<BR>\n" );
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document.write( "You get 100 - 80 = 200 which becomes 120 = 200 which is way too low.\r<BR>\n" );
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document.write( "Double L to 20.\r<BR>\n" );
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document.write( "You get 400 - 160 = 200 which becomes 240 = 200 which is a little too high.\r<BR>\n" );
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document.write( "Since L = 20 was a lot closer to 200 then L = 10, your answer will probably be much closer to 20 than to 10.\r<BR>\n" );
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document.write( "Try 18 for L to get 324 - 144 = 200 which becomes 180 = 200.   Still a little too low.\r<BR>\n" );
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document.write( "Try 19 for L to get 361 - 152 = 200 which becomes 209 = 200.   Just a little too high.\r<BR>\n" );
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document.write( "Your answer is going to be between 18 and 19.\r<BR>\n" );
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document.write( "It will probably be closer to 19 rather than 18 because 19 was only 9 off from 200 while 18 was 20 off.\r<BR>\n" );
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document.write( "Try 18.8 for L to get 203.04 = 200 which is a little too high.\r<BR>\n" );
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document.write( "Try 18.7 for L to get 200.09 = 200 which is very close, but still too high.\r<BR>\n" );
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document.write( "Since it's still too high, you would probably go down to 18.6 to get 197.16 = 200 which means you're too low.\r<BR>\n" );
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document.write( "Your answer will be between 18.6 and 18.7 and will probably be closer to 18.7 because 18.7 was .09 off while 18.6 was 2.84 off.\r<BR>\n" );
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document.write( "You would probably then try 18.68 to get 199.5024 = 200 which is still closer but still too low.\r<BR>\n" );
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document.write( "You would probably then try 18.69 to get 199.7961 which is even closer to 200 but still low.\r<BR>\n" );
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document.write( "Since 18.69 is too low and 18.7 is too high, then the answer has to be between 18.69 and 18.7.\r<BR>\n" );
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document.write( "since 18.7 is .09 too high and 18.69 is .2039 too low, then the answer is probably going to be closer to 18.7 than 18.69.\r<BR>\n" );
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document.write( "My guess is your answer will be close to 18.697.   That would get you 200.001809 which is very close.\r<BR>\n" );
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document.write( "We could narrow it down further, but this looks close enough.\r<BR>\n" );
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document.write( "If you solved the quadratic equation of L^2 - 8*l - 200 = 0, you would get:\r<BR>\n" );
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document.write( "L = 18.69693846\r<BR>\n" );
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document.write( "We did pretty good through guess and check.\r<BR>\n" );
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document.write( "Once you know L, it's a simple matter to find W.\r<BR>\n" );
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document.write( "W = L - 8\r<BR>\n" );
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document.write( "L = 18.69693846 (the real answer, not the one we guessed at).<BR>\n" );
document.write( "W = 10.69693846\r<BR>\n" );
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document.write( "L * W = 18.69693846 * 10.69693846 = 200\r<BR>\n" );
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document.write( "Using our guess, we would have gotten:\r<BR>\n" );
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document.write( "L = 18.697 and W = L - 8 = 10.697\r<BR>\n" );
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document.write( "L * W = 200 would become 18.697 * 10.697 = 200.001809.\r<BR>\n" );
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document.write( "Not too shabby.\r<BR>\n" );
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document.write( "The guess and check method is used in a lot of cases where a formula is not possible.\r<BR>\n" );
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document.write( "It lends itself very well to computers which can do the calculations extremely fast.\r<BR>\n" );
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document.write( "You can also use interpolation to narrow your next guess down closer.\r<BR>\n" );
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document.write( "I did some of that by recognizing that the answer with L = 10 was much further away than the answer with L = 20 which led me to conclude that the answer was probably going to be somewhere between 18.8 and 20.\r<BR>\n" );
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document.write( "I was a little off but not too far since the real answer was 18.697.\r<BR>\n" );
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document.write( "With computers, the interpolation method would be by formula rather than guess work.\r<BR>\n" );
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document.write( "My third guess would have been x = 18.1818181818 if I used an interpolation formula.\r<BR>\n" );
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document.write( "This wasn't too far off from my third guess of 18.\r<BR>\n" );
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document.write( "Guess and Check is a laborious process that computers do extremely well.\r<BR>\n" );
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document.write( "It's used in a lot of cases where solving by formula directly is just not possible.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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