# Volume of Set S in Terms of L, W, & H: Problem of the Day

• Yoda13
In summary, the conversation is discussing how to express the volume of a set of points that are within a certain distance of a solid box. This volume can be calculated by adding the surface area of the box (2lw + 2hw + 2lh) to the volume of a unit sphere (4/3*pi) multiplied by the height of the box (H) and the length (L) and width (W) of the box. The conversation also mentions the difficulty in calculating the volume for points inside the box at a specific distance, and the possibility of constructing a function to find this volume. Finally, the conversation clarifies the presence of additional terms (l(pi) + h(pi) + w(pi)) due to the

#### Yoda13

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. Have fun.

lwh+2lw+2hw+2lh+h(pi)+l(pi)+w(pi)

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Don't forget the corners: pasted together they constitute a unit-sphere, so there is an extra contribution of 4/3 pi to the volume of S:

$$L W H+2L W+2H W+2L H+\pi H+\pi L+\pi W + \frac{4}{3}\pi$$

Ok thanks. Now I know what you guys were talking about. I though he meant a point INSIDE the box say (h,j,k). I think that's a little more difficult: what is the volume of the solid formed by all points inside the box that are a distance of 1 unit away from the point (say for a box 2x2x2)? Are there points in the box that yield a miniumum volume? Is it at the corners? So conceivably, we could construct a function of 3 variables: f(x,y,z), which yields this volume as a function of position in the box. Really, I think just calculating one such volume would be difficult.

o right, i forgot about the corners, thanks

can someone please explain where the l(pi)+h(pi)+w(pi) came from. sorry I am a physics student and we were given a similar problem, and i just don't really understand. our was for a different shape and different distance but i think this helps.