Question

Let $B$ be a right rectangular prism (box) with edge lengths $1,$ $3,$ and $4,$ together with its interior. For real $r\geq0,$ let $S(r)$ be the set of points in $3$ -dimensional space that lie within a distance $r$ of some point in $B.$ The volume of $S(r)$ can be expressed as $ar^{3} + br^{2} + cr +d,$ where $a,$ $b,$ $c,$ and $d$ are positive real numbers. What is $\dfrac{bc}{ad}?$

$6$

$19$

$24$

$26$

$38$

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Written solution:

Decompose $S(r)$ by where the added volume lies relative to the box.

The original box has volume $d=1\cdot3\cdot4=12$ . The face slabs contribute surface area times $r$ , so $c=2(1\cdot3+1\cdot4+3\cdot4)=38.$

Along each edge is a quarter-cylinder of radius $r$ . The sum of all edge lengths is $4(1+3+4)=32$ , so $b=\frac14\pi\cdot 32=8\pi.$ At the eight corners, the eighth-spheres combine to one full sphere, so $a=\frac43\pi.$

Therefore $\frac{bc}{ad}=\frac{(8\pi)(38)}{(\frac43\pi)(12)}=19.$

Thus, the correct answer is B .

2020 AMC 10B Problem 20

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Problem 20 in Other Years