题目信息
Sprinklers are being installed to water a lawn. Each sprinkler waters in a circle. Can the lawn be watered completely by 4 installed sprinklers?
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.
参考答案及共享解析
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已选答案：
正确答案：
E:Statements (1) and (2) TOGETHER are NOT sufficient.
Geometry Circles; Rectangles
No information is given about the area of the region that can be completely covered by four installed sprinklers; NOT sufficient. No information is given about the area or the shape of the lawn; NOT sufficient.
Given (1) and (2), if the length of the rectangular lawn is sufficiently large, for example if the length is 32 yards and the width is 1 yard, then it is clear that the four sprinklers cannot completely water the lawn. However, if the lawn is in the shape of a square, then it is possible that four sprinklers can completely water the lawn. To see this, we first note that the side length of the square lawn is
yards. To assist with the mathematical details, the figure below shows the square lawn positioned in the standard (x,y) coordinate plane so that the vertices of the lawn are located at (0,0), 
, 
, and 
. The two diagonals of the square, each of length 8, are shown as dashed segments, and the four sprinklers are at the four marked points located at the midpoints of the left and right halves of the diagonals. For example, one of the sprinklers is located at the point 
. Using the distance formula, it is straightforward to show that a circle centered at with radius 2 passes through each of the points (0,0), 
, 
, and 
. Therefore, the interior of this circle covers the lower left square portion of the square lawn—that is, the square portion having vertices (0,0), 
, 
, and 
. Hence, the four sprinklers together, when located as described above, can completely water the square lawn. Therefore, it is possible that the lawn cannot be completely watered by the four sprinklers, and it is possible that the lawn can be completely watered by the four sprinklers.
The correct answer is E;both statements together are still not sufficient.
No information is given about the area of the region that can be completely covered by four installed sprinklers; NOT sufficient. No information is given about the area or the shape of the lawn; NOT sufficient.
Given (1) and (2), if the length of the rectangular lawn is sufficiently large, for example if the length is 32 yards and the width is 1 yard, then it is clear that the four sprinklers cannot completely water the lawn. However, if the lawn is in the shape of a square, then it is possible that four sprinklers can completely water the lawn. To see this, we first note that the side length of the square lawn is
yards. To assist with the mathematical details, the figure below shows the square lawn positioned in the standard (x,y) coordinate plane so that the vertices of the lawn are located at (0,0), , , and . The two diagonals of the square, each of length 8, are shown as dashed segments, and the four sprinklers are at the four marked points located at the midpoints of the left and right halves of the diagonals. For example, one of the sprinklers is located at the point . Using the distance formula, it is straightforward to show that a circle centered at with radius 2 passes through each of the points (0,0), , , and . Therefore, the interior of this circle covers the lower left square portion of the square lawn—that is, the square portion having vertices (0,0), , , and . Hence, the four sprinklers together, when located as described above, can completely water the square lawn. Therefore, it is possible that the lawn cannot be completely watered by the four sprinklers, and it is possible that the lawn can be completely watered by the four sprinklers.The correct answer is E;both statements together are still not sufficient.
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