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
Video Solution
~savannahsolver
Video Solution
~Charles3829
Video Solution by TheBeautyofMath
https://youtu.be/o98vGHAUYjM?t=158
~IceMatrix
Video Solution (HOW TO THINK CREATIVELY!!!)
~Education, the Study of Everything
See Also
2021 Fall AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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