Difference between revisions of "2021 Fall AMC 10A Problems/Problem 3"
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<math>\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }7</math> | <math>\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }7</math> | ||
− | == Solution == | + | == Solution 1 == |
A sphere with radius <math>2</math> has volume <math>\frac {32\pi}{3}</math>. A cube with side length <math>6</math> has volume <math>216</math>. If <math>\pi</math> was <math>3</math>, it would fit 6.75 times inside. Since <math>\pi</math> is approximately <math>5</math>% larger than <math>3</math>, it is safe to assume that the <math>3</math> balls of clay can fit <math>6</math> times inside. Therefore, our answer is <math>\boxed {(D)6}</math>. | A sphere with radius <math>2</math> has volume <math>\frac {32\pi}{3}</math>. A cube with side length <math>6</math> has volume <math>216</math>. If <math>\pi</math> was <math>3</math>, it would fit 6.75 times inside. Since <math>\pi</math> is approximately <math>5</math>% larger than <math>3</math>, it is safe to assume that the <math>3</math> balls of clay can fit <math>6</math> times inside. Therefore, our answer is <math>\boxed {(D)6}</math>. | ||
~Arcticturn | ~Arcticturn | ||
+ | |||
+ | == Solution 2 == | ||
+ | |||
+ | The volume of the cube is <math>6^3 = 216.</math> The volume of the sphere is <math>\frac{4}{3} \pi r^3 = \frac{4}{3} \pi \cdot 8 = \frac{32}{3} \pi.</math> Because the balls can be compressed but not reshaped, the greatest number of balls that can fit inside the cube is <math> \left\lfloor \frac{216}{\frac{32}{3}\pi} \right\rfloor = 6 .</math> Thus, the answer is <math>\boxed{\textbf{(D)}.}</math> | ||
+ | |||
+ | ~NH14 | ||
==See Also== | ==See Also== | ||
{{AMC10 box|year=2021 Fall|ab=A|num-b=2|num-a=4}} | {{AMC10 box|year=2021 Fall|ab=A|num-b=2|num-a=4}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 20:27, 22 November 2021
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
A sphere with radius has volume
. A cube with side length
has volume
. If
was
, it would fit 6.75 times inside. Since
is approximately
% larger than
, it is safe to assume that the
balls of clay can fit
times inside. Therefore, our answer is
.
~Arcticturn
Solution 2
The volume of the cube is The volume of the sphere is
Because the balls can be compressed but not reshaped, the greatest number of balls that can fit inside the cube is
Thus, the answer is
~NH14
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 |
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.