2. 外语
  3. The U.S. Defense Department has decided that the Pentagon is an obsolete building and that it must be replaced with an upgraded

The U.S. Defense Department has decided that the Pentagon is an obsolete building and that it must be replaced with an upgraded

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问题 The U.S. Defense Department has decided that the Pentagon is an obsolete building and that it must be replaced with an upgraded version: the Hexagon. The Secretary of Defense wants a building that is exactly 70 feet high and 200 feet on a side, and that has a hexagonal bull’s-eye cutout in the center (somewhat like the current one) that is 50 feet on a side. What will be the volume of the new building in cubic feet?
选项 A、3,937,500 cubic feet
B、15,750 cubic feet
C、 
D、 
E、 
答案E
解析 This is very definitely a multi-step problem. First, you need to recognize that we’re dealing with nothing more than an oddly-shaped cylinder: the base is a regular polygon, so by dividing it into a number of manageable pieces, we can calculate its area. However, before we do that, we’ll need to figure out how to treat that hole in the middle of the six-sided doughnut. The easiest thing to do is to think about it as "negative area"--to calculate its area, and then subtract that area from the area of the larger, filled-in shape to find the area of the region around it. The following diagram below shows the three steps:
   Next, we’ll need to calculate the areas of those two different triangles. If you happen to remember the formula for the area of an equilateral triangle, great; skip to that step. If not, you’ll need to derive it. That’s not that hard: using the diagram that follows, you can divide the equilateral triangle into two right triangles, each with height h, hypotenuse a (there go those shifting variable names again), and base a/2. Since we don’t know what h is, it would be nice to state that in terms of a as well. We’ll do that first, by relying on the Pythagorean Theorem:

   Step 4: Calculate the area of the triangle
   The area of the entire equilateral triangle is just twice the area of the right "half triangle" on either side:

   Six of these form the whole of the large (outer) hexagon, so that area is:

   Six of the smaller triangles form the whole of the hole in the middle. We’ll need a different variable for these, since we’re talking about a different a at this point. We can define these as the outer ao and inner ai sides:
   The area of the Hexagon (minus the bull’s eye)=
   Then, the height is a uniform 70 feet all around, so finding the volume is just a matter of multiplying through by that height:
   Of course, that doesn’t look much like the answer choices, so we’ll need to substitute the values for the variables:
   If you looked over the answer choices, you will know that  is an acceptable component of your answer; the thrust of this question is geometry, so the GMAT would not make you calculate a square root as part of the problem. The volume corresponds with answer choice E.
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