How do I express the volume of a set in terms of length, width, and height?

[stevo13]

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.

 

[EnumaElish]

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.

 

[stevo13]

Thanks guys for your help. Very clear explanations.