2021 Fall AMC 10A Problems/Problem 3
Contents
Problem
What is the maximum number of balls of clay of radius that can completely fit inside a cube of side length
assuming the balls can be reshaped but not compressed before they are packed in the cube?
Solution 1 (Inequality)
The volume of the cube is and the volume of a clay ball is
Since the balls can be reshaped but not compressed, the maximum number of balls that can completely fit inside a cube is
Approximating with
we have
or
We simplify to get
from which
~NH14 ~MRENTHUSIASM
Solution 2 (Inequality)
As shown in Solution 1, we conclude that the maximum number of balls that can completely fit inside a cube is
By an underestimation we have
or
By an overestimation we have
or
Together, we get
from which
~MRENTHUSIASM
Solution 3 (Approximation)
As shown in Solution 1, we conclude that the maximum number of balls that can completely fit inside a cube is
Approximating with we have
Since
is about
greater than
it is safe to claim that
~Arcticturn ~MRENTHUSIASM
See Also
2021 Fall AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
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