document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=102728>Question 102728</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help this is really hard!!<BR>\n" );
document.write( "The answer is Length=16 meters;width=7 meters. How will I get it??<BR>\n" );
document.write( "Problem: There is a rectangle whose perimeter is 18 centimeters. If its length is decreased by 5 centimeters and its width is increased by 12 centimeters, its area is doubled. Find 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 #74714 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=74714\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You've given the wrong answers for this problem. The first clue is that the answers you<BR>\n" );
document.write( "provided for Length and Width are in meters while the dimensions in your problem are in<BR>\n" );
document.write( "centimeters. Another check is that the perimeter of the rectangle is supposed to be 18 cm.<BR>\n" );
document.write( "However, your answers of Length = 16 and Width = 7 would give a perimeter of 16 + 7 + 16 + 7<BR>\n" );
document.write( "which totals 46.<BR>\n" );
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document.write( "So let's ignore your answer and work the problem using the information given.<BR>\n" );
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document.write( "Begin by calling L the original length of the rectangle and calling W the original width. <BR>\n" );
document.write( "The perimeter of the rectangle is the distance around it, and this is L + W + L + W which<BR>\n" );
document.write( "combines to 2L + 2W. You are told that this perimeter equals 18 cm. So we can write the<BR>\n" );
document.write( "equation:<BR>\n" );
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document.write( "2L + 2W = 18<BR>\n" );
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document.write( "and we can reduce this a little by dividing both sides (all terms) by 2 to get:<BR>\n" );
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document.write( "L + W = 9<BR>\n" );
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document.write( "Let's now solve this for one of the quantities in terms of the other. Let's say we solve<BR>\n" );
document.write( "for L in terms of W. Do this by subtracting W from both sides of this equation to get:<BR>\n" );
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document.write( "L = 9 - W<BR>\n" );
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document.write( "One final step in this first part. Let's find the area of the original rectangle. We<BR>\n" );
document.write( "know that the area (A) of a rectangle is found by multiplying the length L by the width W.<BR>\n" );
document.write( "Therefore we can write the equation:<BR>\n" );
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document.write( "A = L*W<BR>\n" );
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document.write( "But we found that for this rectangle L is equal to 9 - W. Therefore, we can substitute<BR>\n" );
document.write( "9 - W for L in the area equation and we get that the area of the original rectangle <BR>\n" );
document.write( "is:<BR>\n" );
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document.write( "A = L*W = (9 - W)*W = 9W - W^2<BR>\n" );
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document.write( "Now to the second part of the problem. If we decrease the original length by 5 cm, the<BR>\n" );
document.write( "new length is L - 5. Then if we increase the original width by 12 cm the new width is<BR>\n" );
document.write( "W + 12. Therefore, the area of the new rectangle is found by multiplying the new length by<BR>\n" );
document.write( "the new width which can be written as:<BR>\n" );
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document.write( "New Area = (L - 5)(W + 12)<BR>\n" );
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document.write( "But recall that L is equal to 9 - W, so replace L by 9 - W in the New Area equation to<BR>\n" );
document.write( "get:<BR>\n" );
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document.write( "New Area = (9 - W - 5)(W + 12)<BR>\n" );
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document.write( "Combine the numbers 9 and -5 in the first set of parentheses and the equation reduces to:<BR>\n" );
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document.write( "New Area = (4 - W)(W + 12)<BR>\n" );
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document.write( "Multiply out the right side and the equation becomes:<BR>\n" );
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document.write( "New Area = 4W + 48 - W^2 - 12W <BR>\n" );
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document.write( "and combining like terms 4W and - 12W the equation reduces to:<BR>\n" );
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document.write( "New Area = -W^2 - 8W + 48<BR>\n" );
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document.write( "The problem tells you that this new area equals twice the old area. Recall that earlier we<BR>\n" );
document.write( "found that the old area was 9W - W^2 so twice the old area would be 18W - 2W^2. Now we<BR>\n" );
document.write( "can write the equation that the new area (-W^2 - 8W + 48) equals twice the old area (18W - 2W^2) ...<BR>\n" );
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document.write( "-W^2 - 8W + 48 = 18W - 2W^2<BR>\n" );
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document.write( "Let's get rid of the two terms on the right side. First get rid of the - 2W^2 by adding 2W^2 <BR>\n" );
document.write( "to both sides to get:<BR>\n" );
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document.write( "W^2 - 8W + 48 = 18W<BR>\n" );
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document.write( "Next get rid of the 18W on the right side by subtracting 18W from both sides:<BR>\n" );
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document.write( "W^2 - 26W + 48 = 0<BR>\n" );
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document.write( "Next factor the left side:<BR>\n" );
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document.write( "(W - 24)(W - 2) = 0<BR>\n" );
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document.write( "Note that this equation will be true if either of the factors on the left side is zero because<BR>\n" );
document.write( "a multiplication by zero on the left side will make the left side zero and therefore <BR>\n" );
document.write( "equal to the right side.<BR>\n" );
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document.write( "By setting the first factor (W - 24) equal to zero and solving for W we get that W = +24 is<BR>\n" );
document.write( "a possible answer. But is it really??? If the original width is 24, then the original<BR>\n" );
document.write( "perimeter has to be much bigger than 18. So toss out the answer W = 24.<BR>\n" );
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document.write( "By setting the second factor equal to zero we find that W = +2. That looks a lot better. Going<BR>\n" );
document.write( "way back to the beginning of our work we had found that L = 9 - W and if W is 2, then <BR>\n" );
document.write( "L = 9 - 2 = 7.<BR>\n" );
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document.write( "So it looks as if the dimensions of the original rectangle is 7 cm long and 2 cm wide.<BR>\n" );
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document.write( "Let's check. The perimeter of the original rectangle would be 7 + 2 + 7 + 2 = 18 cm. That's OK.<BR>\n" );
document.write( "The Area of the original rectangle is 7 times 2 or 14 square cm.<BR>\n" );
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document.write( "Now decrease the original length by 5 cm and you get 2 cm. Then increase the original<BR>\n" );
document.write( "width by 12 cm and you get 2 + 12 = 14 cm. So the new figure has dimensions of 2 cm by 14 cm.<BR>\n" );
document.write( "The area of this new figure is the product of 2 cm by 14 cm which equals 28 square cm.<BR>\n" );
document.write( "And this area (28) is twice the square cm of the original rectangle (14). So everything<BR>\n" );
document.write( "works out and our answers of 2 cm for the original width and 7 cm for the original <BR>\n" );
document.write( "length are correct.<BR>\n" );
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document.write( "A lot of work ... hope this helps you to see your way through this problem.<BR>\n" );
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