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  1. Aug 14, 2005 #1
    Hi, guys. In a previous post this problem was given: Let B be a solid box with length L, width W, and height H. Let S be the set of all points that are a distance at most 1 from some point of B. Express the volume of S in terms of L,W, and H. How do you attack this problem? I'm not sure where to start or how to relate L,W, and H to anything. Thanks for your input.
  2. jcsd
  3. Aug 14, 2005 #2
    first picture the cube with length l, width w, and height h. Now, everypoint that lies within 1 of this cube becomes the new shape that we need to find a volume for. I started by breaking down the new cube into the old cube pluse 6 new sides, 12 new edges, and 4 new corners. Lets start with the sides. Each side is 1 unit wide, long or high, depending on what side you decide to start with. so you add (*=times)2*l*w+2*l*h+2*w*h to the volume of the original cube. Since there are six sides, you must add each side twice, hence 2*. Now you have the original cube plus the sides of your new shape. Next, we move on to the edges, l,w, and h. The edge of the new shape will look like 1/4 of a cylinder, or, 1/4 the area of a circle *l,w, or h. Since the area of a circle is pi*r^2, the volume of each new edge type shape is pi/4*l,w, or h, depending. Since there are 4 edges with l, 4 with w, and 4 for h, we can cancel the denominator to come up with pi*l+pi*w+pi*h. Finally, the corners. pasted together they constitute a unit-sphere, so there is an extra contribution of 4/3 pi to the volume of S:
    lwh+2lw+2lh+2hw+pi(h)+ pi(l)+pi(w)+4/3Pi
  4. Aug 14, 2005 #3


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    Simplify to a cube with side length L. To simplify even further, think of a 2-D "cube" which is a square. How would you calculate the AREA of a surface that embraces the square and has its boundary set at 1 unit distance from the square? I think the tricky part is the corners; other than the corners it is straightforward.

    Back to the solid cube. Here is a mental image that helped me to think about the problem. Suppose the cube is made of hard plastic. You are to stick some needles in the middle of each surface, then on the middle of each side, then at each corner of the cube. Needles are designed so that the portion protruding outside the cube is exactly one unit in length. After needling is finished, you are to wrap some cellophane film over the needles. The volume inside the cellophane wrap is the volume that the problem is asking.

    I promise to write again if I can think of anything even brighter. :smile:
  5. Aug 15, 2005 #4
    Thanks guys for your help. Very clear explanations.
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