document.write( "Question 1097200: I ONLY NEED TO KNOW PART C
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document.write( "5a. The sides of a square are 16 cm in length. The midpoints of the sides of this square are joined to form a new square and four triangles. The process is repeated twice.
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document.write( "Let Xn denote the length of one of the equal sides of each new triangle.
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document.write( "Let An denote the area of each new triangle.
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document.write( "The following table gives the values of Xn and An, for 1 less than or equal to n less than or equal to 3.\r
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document.write( "N 1 2 3
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document.write( "Xn 8 __ 4
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document.write( "An 32 16 __
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document.write( "(First blank I got 6, second blank I got 8)\r
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document.write( "5b. The process described above is repeated. Find A6.
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document.write( "A6 = 8\r
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document.write( "5c. Consider an initial square of side length k cm. The process describes above is repeated indefinitely. The total area of the shaded regions is k cm^2. Find the value of k. \n" );
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Algebra.Com's Answer #711616 by KMST(5328)![]() ![]() You can put this solution on YOUR website! You need more help than that. \n" ); document.write( " \n" ); document.write( "You are half right on question 5a. \n" ); document.write( "As you draw each new square, \n" ); document.write( "the new square area is half of the area of the previous square. \n" ); document.write( " \n" ); document.write( "The square can be split into 4 more triangles, \n" ); document.write( "as I did with those green lines, \n" ); document.write( "and the other half of the previous square area is made of the 4 new triangles. \n" ); document.write( "The areas of squares and triangles form a geometric sequence with common ratio 1/2 \n" ); document.write( "So, the areas are 32,16,8,4,2,1,1/2,1/4, and so on. \n" ); document.write( "The terms of that sequence are called \n" ); document.write( "The second blank (area of the third triangle) is indeed \n" ); document.write( " \n" ); document.write( "Because areas are made by multiplying lengths of sides and a convenient factor, \n" ); document.write( "lengths of sides form a geometric sequence with common ratio \n" ); document.write( " \n" ); document.write( "So, the sides are 8,4sqrt(2),4,2sqrt(2),2,sqrt(2),1, and so on. \n" ); document.write( "The terms of that sequence of lengths are called \n" ); document.write( "The first blank (side of the second triangle) is not 6, but \n" ); document.write( " \n" ); document.write( "5b. You may have misinterpreted that question. \n" ); document.write( "The question asks for \n" ); document.write( "I do not know how you came up with your answer, but the sequence of areas is \n" ); document.write( "32,16,8,4,2,1,1/2,1/4, and so on, and the 6th term there is \n" ); document.write( " \n" ); document.write( " \n" ); document.write( "5c. You did not include a picture so that we could figure out what area was shaded in the picture that came with your question. \n" ); document.write( "However, I did an internet search using a phrase with your question. \n" ); document.write( "I found the question to show one shaded triangle from the each step, \n" ); document.write( "all adjacent to the triangles in the steps before and after, \n" ); document.write( "forming a clockwise spiral. \n" ); document.write( "The area shaded after an infinite number of steps is the sum of the \n" ); document.write( "32+16+8+4+2+1+....=64. \n" ); document.write( "If you had to \"show your work\" the teacher may expect you to say that \n" ); document.write( "the areas form a geometric sequence with common ratio \n" ); document.write( "and because that ratio is less than 1, the series (the sum) converges to \n" ); document.write( " \n" ); document.write( "However, a fifth grader (or someone smarter than that) \n" ); document.write( "would realize that each sum gets closer to 64, \n" ); document.write( "and that what it needs to get to 64 is exactly the last term added. \n" ); document.write( "Adding we would never get to 64, \n" ); document.write( "because we would need to add infinite terms, \n" ); document.write( "and our lives are finite, \n" ); document.write( "but the sum to infinity is 64, \n" ); document.write( "because we see that \n" ); document.write( "if there was such a real number, would be zero. \n" ); document.write( "If the first square has side length \n" ); document.write( "the area of that square would be \n" ); document.write( " \n" ); document.write( "Then the sum of infinite triangles would be \n" ); document.write( "If that is equal to \n" ); document.write( " \n" ); document.write( "and there really was an initial square, \n" ); document.write( "we can divide both sides by \n" ); document.write( " |