What is the volume of the set of points within 1 unit of a solid box?

[Chris L T521]

Thanks again to those who participated in last week's POTW! Here's this week's problem!

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Problem: 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 $H$.

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

[Chris L T521]

This week's problem was correctly answered by magneto, MarkFL, Opalg, and Pranav. You can find magneto's solution below.

[sp]We will construct the "meta-box" and add the volume.

(a) The volume of the box is $LWH$.

(b) We can extend each of the surface of the box by 1. This creates two blocks of boxes of volume $LW$, $LH$, and $WH$ each. So the total volume would be $2(LW+LH+WH)$.

(c) There are 8 corners each is of $\frac{1}{8}$ of a sphere of radius $1$, so the total volume would be $\frac{4}{3} \pi$.

(d) There are 12 quarter-cylinder, 4 for each dimension of the box. These cylinder's have height of the side of the box, and radius of $1$,
so the total volume is given by $\pi(L+W+H)$.

Therefore, the total volume is $LWH + 2(LW+LH+WH) + \pi (\frac{4}{3} + L + W + H)$.[/sp]