Volumes of Spheres around a Box

[cmdro]

Homework Statement

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.

Homework Equations

4/3(pi)r

The Attempt at a Solution

I have gotten to LWH+2LH+2LW+2HW+4/3(pi) and I think I am missing something involving a pi, but I do not know how to get it right. The LHW is the box volume. The 2 times the areas are to compensate for the encasing on all sides. The 4/3(pi) is for the unit sphere made in the corners. Can you help me?

 

[Dick]

I hope you mean (4/3)pi*r^3. But that's a good start. You are missing some quarter cylinder regions along the edges.

 

[cmdro]

Sorry. That is what I meant. I'm just bad at typing. I apologize. So the cylinder is given by (pi)(r)(height) correct? r=1. So pi(l) +pi(w) +pi(h)?

 

[Dick]

The cylinder is pi*r^2*h. You keep getting the exponent of r wrong. Since r=1 it doesn't show up in the answer, but yes, I think that's correct.

 

[cmdro]

Thank you sooooo much! And sorry. I am trying to type fast and keep forgetting. I write them down right, but type them wrong. But thanks for correcting me, making sure I'm right! Thanks!