document.write( "Question 1120369: A rectangle has an area of 247 cm2 and a perimeter of 64 cm. What are its dimensions?
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Algebra.Com's Answer #736102 by ikleyn(52787) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "I present here three major methods of solving this problem (and solving other problems of the same type).\r
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\n" ); document.write( "\n" ); document.write( "1-st method\r
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\n" ); document.write( "\n" ); document.write( "First method is to create a corresponding quadratic equation and then solve it using the quadratic formula.\r
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document.write( "Let  x  be the length of the rectangle, in centimeters, and \r\n" );
document.write( "let  y  be its width.\r\n" );
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document.write( "Then 2x + 2y = 64.\r\n" );
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document.write( "Hence,  2y = 64 - 2x,  and  y =  = 32 - x.\r\n" );
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document.write( "The area of the rectangle is \r\n" );
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document.write( "x*y = x*(32-x) = - x^2 + 32x.\r\n" );
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document.write( "It is equal to 247 cm^2, which gives you an equation\r\n" );
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document.write( "-x^2 + 32x = 247,   or\r\n" );
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document.write( "x^2 - 32x + 247 = 0.      (1)\r\n" );
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document.write( "Apply the quadratic formula \r\n" );
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document.write( " =  =  =  = .\r\n" );
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document.write( "You have two roots:   = 16 + 3 = 19  and   = 16 - 3 = 13.\r\n" );
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document.write( "Usually, the greater dimension is called the length.\r\n" );
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document.write( "So, in this problem the length is 19 centimeters.\r\n" );
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document.write( "Then the width is  32-19 = 13 centimeters.\r\n" );
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document.write( "Answer.  The dimensions of the rectangle are  19 centimeters and  13 centimeters.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Regarding this method, I'd like specially to highlight that the quadratic formula works for any quadratic equation.\r
\n" ); document.write( "\n" ); document.write( "It works like an army tank and provides a solution to any quadratic equation.\r
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\n" ); document.write( "\n" ); document.write( "2-nd method\r
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\n" ); document.write( "\n" ); document.write( "Second method is to create a correcponding quadratic equation and then solve it using factoring its left side.\r
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document.write( "Surely, the equation is the same as (1)\r\n" );
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document.write( "x^2 - 32x + 247 = 0,      (1)\r\n" );
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document.write( "and the method of obtaining this equation remains the same as in the Solution 1.\r\n" );
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document.write( "Now we want to factor the left side in the form\r\n" );
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document.write( "(x - ?)*(x - ?) = 0.       (2)\r\n" );
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document.write( "and now your goal is to find the numbers to put them instead of the question marks in order for to get the same equation (1).\r\n" );
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document.write( "It not very easy task. Your first wish is to try integer divisors of the number 247.\r\n" );
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document.write( "After some trials you will find them as 13 and 19.\r\n" );
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document.write( "Notice that their product is  13*19 = 247  and their sum is  13+19 = 32,\r\n" );
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document.write( "so (2) becomes equivalent to equation (1).\r\n" );
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document.write( "Now you get the same answer  x= 19 as in the Solution 1.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Regarding this method, I'd like specially to highlight that it works good and fast when the coefficients of your original equation
\n" ); document.write( "are small and good enough to that extent that you can mentally find the required combination quickly.\r
\n" ); document.write( "\n" ); document.write( "Another situation when it works good is when you know the solution in advance.\r
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\n" ); document.write( "\n" ); document.write( "3-rd method\r
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\n" ); document.write( "\n" ); document.write( "It is most beatiful method.\r
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document.write( "You are given that the perimeter of the rectangle is 64 cm;  hence, the sum of the length and the width is one half of that:\r\n" );
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document.write( "x + y = 32.\r\n" );
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document.write( "Then the average of the length and the width is one half of 32, i.e. 16.\r\n" );
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document.write( "It is clear that the values of x and y are remoted in the same value/distance \"u\" from 16, so we can write\r\n" );
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document.write( "x = 16 + u,\r\n" );
document.write( "y = 16 - u.\r\n" );
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document.write( "Then the area is  xy = (16+u)*(16-u) = , and it is equal to 247, according to the condition.\r\n" );
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document.write( "Hence,   = 247,  which gives   = 256 - 247 = 9,  and then  u =  = 3.\r\n" );
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document.write( "Thus the length is  x = 16 + u = 16 + 3 = 19,\r\n" );
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document.write( "and  the width  is  x = 16 - u = 16 - 3 = 13.\r\n" );
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document.write( "We get the same answer quickly and mentally.\r\n" );
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\n" ); document.write( "\n" ); document.write( "You may find the third method in some recreational books.\r
\n" ); document.write( "\n" ); document.write( "Sometimes in math circles the teachers explain it to students.\r
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\n" ); document.write( "\n" ); document.write( "In the school Math, the standard method of solution is to reduce the problem to a quadratic equation\r
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\n" ); document.write( "\n" ); document.write( "I presented the third method here to make your horizon wider.\r
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\n" ); document.write( "\n" ); document.write( "If you demonstrate this method in the class, your teacher and your classmates will be impressed !\r
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\n" ); document.write( "\n" ); document.write( "So, it was as if you visited a Math circle today . . . \r
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\n" ); document.write( "\n" ); document.write( "To see other solved similar problems using the third method, look into the lesson\r
\n" ); document.write( "\n" ); document.write( " - HOW TO solve the problem on quadratic equation mentally and avoid boring calculations \r
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\n" ); document.write( "\n" ); document.write( "Also, you have this free of charge online textbook in ALGEBRA-I in this site\r
\n" ); document.write( "\n" ); document.write( " - ALGEBRA-I - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lesson is the part of this textbook under the topic \"Quadratic equations\". \r
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\n" ); document.write( "\n" ); document.write( "Save the link to this online textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-I
\n" ); document.write( "https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "to your archive and use it when it is needed.\r
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