document.write( "Question 1097200: I ONLY NEED TO KNOW PART C
\n" ); 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.
\n" ); document.write( "Let Xn denote the length of one of the equal sides of each new triangle.
\n" ); document.write( "Let An denote the area of each new triangle.
\n" ); 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
\n" ); document.write( "\n" ); document.write( "N 1 2 3
\n" ); document.write( "Xn 8 __ 4
\n" ); document.write( "An 32 16 __
\n" ); document.write( "(First blank I got 6, second blank I got 8)\r
\n" ); document.write( "\n" ); document.write( "5b. The process described above is repeated. Find A6.
\n" ); document.write( "A6 = 8\r
\n" ); document.write( "\n" ); 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.
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Algebra.Com's Answer #711616 by KMST(5328)\"\" \"About 
You can put this solution on YOUR website!
You need more help than that.
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\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.
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\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 \"A%5Bn%5D\"
\n" ); document.write( "The second blank (area of the third triangle) is indeed \"A%5B3%5D=8\" .
\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( "\"sqrt%281%2F2%29=sqrt%282%29%2F2\" .
\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 \"X%5Bn%5D\" .
\n" ); document.write( "The first blank (side of the second triangle) is not 6, but \"X%5B2%5D=4sqrt%282%29\" .
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\n" ); document.write( "5b. You may have misinterpreted that question.
\n" ); document.write( "The question asks for \"A%5B6%5D\" , the 6th term of the sequence of areas that I listed above.
\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( "\"A%5B6%5D=1\" .
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\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 \"A%5Bn%5D\" terms from start to infinity:
\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 \"r=1%2F2\" .
\n" ); document.write( "and because that ratio is less than 1, the series (the sum) converges to
\n" ); document.write( "\"S=A%5B1%5D%2F%281-r%29\"\"%22=%22\"\"32%2F%281-1%2F2%29\"\"%22=%22\"\"32%2F%281%2F2%29\"\"%22=%22\"\"32%2A2=64\" .
\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 \"A%5Binfinity%5D\" ,
\n" ); document.write( "if there was such a real number, would be zero.
\n" ); document.write( "If the first square has side length \"k\" ,
\n" ); document.write( "the area of that square would be \"k%5E2\" and the area of each of the first set of tr triangles would be
\n" ); document.write( "\"A%5B1%5D=k%5E2%2F8\" .
\n" ); document.write( "Then the sum of infinite triangles would be \"2A%5B1%5D=k%5E2%2F4\" .
\n" ); document.write( "If that is equal to \"k\" ,
\n" ); document.write( "\"k%5E2%2F4=k\" , meaning that if \"k%3C%3E0\" ,
\n" ); document.write( "and there really was an initial square,
\n" ); document.write( "we can divide both sides by \"k\" to get
\n" ); document.write( "\"k%2F4=1\" and \"highlight%28k=4%29\" .
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