单选题
When opened and lying flat, a birthday card is in the shape of a regular hexagon. The card must be folded in half along 1 of its diagonals before being placed in an envelope for mailing. Assuming that the thickness of the folded card will not be an issue, will the birthday card fit inside a rectangular envelope that is 4 inches by 9 inches?(1) Each side of the regular hexagon is 4 inches long. (2) The area of the top surface (which is the same as the area of the bottom surface) of the folded birthday card is less than 36 square inches.
【正确答案】
A
【答案解析】As shown in the figure above, a regular hexagon with sides of length s can be partitioned into six equilateral triangles. Using these triangles, it is possible to determine the length of each diagonal (2s), the height, shown as a dashed line, of each triangle
the area of eachtriangular region
and the area of the hexagonal region
When the birthday card is folded in half along one of the diagonals it has the shape shown below.
(1) Given s = 4, the maximum width of the birthday card is 2s = 8, which is less than the width of the envelope, and its height is 
，which is less than the height of the envelope because
Thus, the birthday card will fit in the envelope; SUFFICIENT.(2) Given that the surface area of the card when folded is less than 36 square inches, it follows that 
< 36, which simplifies to 
. If s = 4, then the birthday card will fit in the envelope, as shown in (1) above However, if s = 5, then s < 
(note that 625 = 54 < 
= 768), but the maximum width of the birthday card will be 2s = 10, and the card will not fit in the envelope; NOT sufficient.The correct answer is A; statement 1 alone is sufficient.
the area of eachtriangular region and the area of the hexagonal regionWhen the birthday card is folded in half along one of the diagonals it has the shape shown below.(1) Given s = 4, the maximum width of the birthday card is 2s = 8, which is less than the width of the envelope, and its height is ，which is less than the height of the envelope becauseThus, the birthday card will fit in the envelope; SUFFICIENT.(2) Given that the surface area of the card when folded is less than 36 square inches, it follows that < 36, which simplifies to . If s = 4, then the birthday card will fit in the envelope, as shown in (1) above However, if s = 5, then s < (note that 625 = 54 < = 768), but the maximum width of the birthday card will be 2s = 10, and the card will not fit in the envelope; NOT sufficient.The correct answer is A; statement 1 alone is sufficient.