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?

$3$

$4$

$5$

$6$

$7$

Answer: D

Concepts:volume sphere estimation

Difficulty rating: 870

Solution:

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

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.

2021 AMC 10A Fall Problem 3

Answer: D

Difficulty rating: 870

Solution:

Problem 3 in Other Years