document.write( "Question 824149: 1)The perimeter of a rectangle is 28 cm. find the range of possible values of the width of the rectangle if the diagonal is less than 10 cm.\r
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Algebra.Com's Answer #496255 by math-vortex(648) $\"\"$ $\"About$
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document.write( "Hi, there--\r\n" );
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document.write( "YOUR PROBLEM:\r\n" );
document.write( "1)The perimeter of a rectangle is 28 cm. find the range of possible values of the width of \r\n" );
document.write( "the rectangle if the diagonal is less than 10 cm. \r\n" );
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document.write( "2)The area of the rectangle is 12 square cm. Find the range of possible values of the width \r\n" );
document.write( "of the rectangle if the diagonal is more than 5 cm.\r\n" );
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document.write( "SOLUTION #1:\r\n" );
document.write( "1)The perimeter of a rectangle is 28 cm. find the range of possible values of the width of \r\n" );
document.write( "the rectangle if the diagonal is less than 10 cm.\r\n" );
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document.write( "Let D be the diagonal of the rectangle.\r\n" );
document.write( "Let W be the width of the rectangle.\r\n" );
document.write( "Let L be the length of the rectangle.\r\n" );
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document.write( "The formula for the perimeter of a rectangle is 2(W+L). We know that the perimeter is 28 cm, \r\n" );
document.write( "so we can write an expression for the length in terms of W.\r\n" );
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document.write( "2(W + L) = 28\r\n" );
document.write( "W + L = 14\r\n" );
document.write( "L = 14 - W\r\n" );
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document.write( "Now consider the diagonal of the rectangle. If we trace the diagonal from the top left corner \r\n" );
document.write( "to the bottom right corner, then up the width and across the length, we see that we have\r\n" );
document.write( "traced a right triangle. We can use the Pythagorean Equation to find the range of possible \r\n" );
document.write( "values for the width.\r\n" );
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document.write( "The Pythagorean Equation is . In our problem,\r\n" );
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document.write( "a = the width, or W\r\n" );
document.write( "b = the length, or 14 - W\r\n" );
document.write( "c = the diagonal which is less than 10 cm.\r\n" );
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document.write( "Solve the inequality by factoring.\r\n" );
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document.write( "Since the the product of these two factors is less than zero, it must be negative. Thus one \r\n" );
document.write( "factor must be negative and the other positive (\"A negative times a positive is a negative.\") \r\n" );
document.write( "We have two cases:\r\n" );
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document.write( "Case I: W - 6 < 0 and W - 8 > 0  OR  Case 2: W- 6 > 0 and W - 8 < 0\r\n" );
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document.write( "Case 1:\r\n" );
document.write( "W - 6 < 0 and W - 8 > 0\r\n" );
document.write( "W < 6 and W > 8\r\n" );
document.write( "Is is not possible for a number to be less than 6 and greater than 8. Case 1 has no solutions.\r\n" );
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document.write( "Case II:\r\n" );
document.write( "W - 6 > 0 and W - 8 < 0\r\n" );
document.write( "W > 6 and W < 8\r\n" );
document.write( "The numbers that are greater than 6 and less than 8 are all the numbers between 6 and 8. \r\n" );
document.write( "Symbolically, we write this as 6\r\n" );
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document.write( "a is the width, or W\r\n" );
document.write( "b is the length, or  12/W\r\n" );
document.write( "c is the diagonal which is greater than 5.\r\n" );
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document.write( "Multiply every term by W^2 to clear the denominator.\r\n" );
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document.write( "Solve by factoring.\r\n" );
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document.write( "We again have several cases to consider.\r\n" );
document.write( "Case 1:  AND  OR\r\n" );
document.write( "Case 2:  AND \r\n" );
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document.write( "In Case 1:\r\n" );
document.write( " AND \r\n" );
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document.write( "Then,\r\n" );
document.write( "[W<4 and W>-4 or W>4 and W<-4] AND  [W>3 and W>-3  or W<3 and W<-3]\r\n" );
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document.write( "So,\r\n" );
document.write( "W<4 and W>-4 means W is between -4 and 4, or -43 and W>-3 implies that W>3\r\n" );
document.write( "W<3 and W<-3 implies that W<-3\r\n" );
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document.write( "Taken together, we have\r\n" );
document.write( "-43 or W<-3]\r\n" );
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document.write( "Both sides of the AND must be true, so W is between -3 and -4 or between 3 and 4.\r\n" );
document.write( "We cannot have a rectangle with a negative width, so W is between 3 and 4.\r\n" );
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document.write( "In Case #2:\r\n" );
document.write( "Case 2:  AND \r\n" );
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document.write( "Then,\r\n" );
document.write( "[W>4 and W>-4 or W<4 and W<-4] AND  [W>3 and W<-3  or W<3 and W>-3]\r\n" );
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document.write( "So,\r\n" );
document.write( "W>4 and W>-4 implies W>4\r\n" );
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document.write( "On the right side, \r\n" );
document.write( "W cannot be greater than 3 and less than -3\r\n" );
document.write( "W<3 and W>-3 implies that W is between -3 and 3.\r\n" );
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document.write( "Both sides of the AND statement must be true.\r\n" );
document.write( "W cannot be greater than 4 and between -3 and 3.\r\n" );
document.write( "Thus Case II is not a solution.\r\n" );
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document.write( "In the context of our problem, the width is between 3 and 4 cm.\r\n" );
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document.write( "WOW! I hope this helps.\r\n" );
document.write( "Ms. Figgy\r\n" );
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