2020 AMC 10B Problem 20

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

Decompose the enlarged solid into the original box, face slabs, edge quarter-cylinders, and corner eighth-spheres

The coefficient of

r^2

comes from one quarter-cylinder along each edge

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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 .

Problem 20 in other years