2. 外语
  3. Patricia purchased x meters of fencing. She originally intended to use all of the fencing to enclose a square region, but later

Patricia purchased x meters of fencing. She originally intended to use all of the fencing to enclose a square region, but later

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问题 Patricia purchased x meters of fencing. She originally intended to use all of the fencing to enclose a square region, but later decided to use all of the fencing to enclose a rectangular region with length y meters greater than its width. In square meters, what is the positive difference between the area of the square region and the area of the rectangular region?
(1) xy=256
(2) y = 4
选项 A、Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B、Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C、BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D、EACH statement ALONE is sufficient.
E、Statements (1) and (2) TOGETHER are NOT sufficient.
答案B
解析 The square’s perimeter is x meters, and thus the square has adjacent sides of length x/4 meters each. Since the rectangle’s perimeter is also x meters, with adjacent side lengths that differ by y meters, it follows that the rectangle’s length is (x/4+y/2) meters (i.e., lengthen two opposite sides of the square by y/2 meters) and the rectangle’s (x/4+y/2)  meters (i.e., shorten the two other opposite sides of the square by y/2 meters).
Alternatively, letting L and W be the length and width, respectively and in meters, of the rectangle, then we can express each of L and Win terms of x and y by algebraically eliminating L and W from the equations 2L + 2W= x and L = W+y.
2L + 2W = x       given
2(W+y) + 2W = x  substitute
L=W+y
W=x/4+y/2      solve for W
L = x/4 + y/2      use L = W+ y
Therefore, in square meters, the area of the square is (x/4) , the area of the rectangle is(x/4+y/2)(x/4-y/2)=(x/4)2-(y/2)2, and the positive difference between these two areas is(y/2).
Determine the value of(y/2).
(1)    Given xy = 256, it is clearly not possible to determine the value of (y/2)2；NOT sufficient.
(2)    Given y = 4, the value of (y/2)2 is equal to 4; SUFFICIENT.
The correct answer is B;
statement 2 alone is sufficient.
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