2021 AMC 10A Fall Problem 3

Question

What is the maximum number of balls of clay with radius

2

that can completely fit inside a cube of side length

6

assuming that the balls can be reshaped but not compressed before they are packed in the cube?

3

4

5

6

7

Compare volumes because the clay can be reshaped

The count is

\left\lfloor \frac{216}{\frac{32\pi}{3}}\right\rfloor

The cube has volume

6^3=216.

One ball of clay has volume

\frac{4}{3}\pi\cdot 2^3=\frac{32\pi}{3}.

Because the clay may be reshaped but not compressed, the maximum number of balls is

\left\lfloor \frac{216}{\frac{32\pi}{3}}\right\rfloor=\left\lfloor\frac{81}{4\pi}\right\rfloor.

Since

12\lt 4\pi\lt 13,

we have

6\lt \frac{81}{4\pi}\lt \frac{81}{12}\lt 7.

Therefore the floor is

6.

Thus, D is the correct answer.

Problem 3 in other years