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 19:27, 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
