Difference between revisions of "2020 AMC 10B Problems/Problem 10"
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A three-quarter sector of a circle of radius <math>4</math> inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches? | A three-quarter sector of a circle of radius <math>4</math> inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches? | ||
| − | + | <asy> | |
draw(Arc((0,0), 4, 0, 270)); | draw(Arc((0,0), 4, 0, 270)); | ||
draw((0,-4)--(0,0)--(4,0)); | draw((0,-4)--(0,0)--(4,0)); | ||
| − | label(" | + | label("$4$", (2,0), S); |
| − | + | </asy> | |
<math>\textbf{(A)}\ 3\pi \sqrt5 \qquad\textbf{(B)}\ 4\pi \sqrt3 \qquad\textbf{(C)}\ 3 \pi \sqrt7 \qquad\textbf{(D)}\ 6\pi \sqrt3 \qquad\textbf{(E)}\ 6\pi \sqrt7</math> | <math>\textbf{(A)}\ 3\pi \sqrt5 \qquad\textbf{(B)}\ 4\pi \sqrt3 \qquad\textbf{(C)}\ 3 \pi \sqrt7 \qquad\textbf{(D)}\ 6\pi \sqrt3 \qquad\textbf{(E)}\ 6\pi \sqrt7</math> | ||
Revision as of 17:27, 7 February 2020
Problem 10
A three-quarter sector of a circle of radius 
inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches?
![[asy] draw(Arc((0,0), 4, 0, 270)); draw((0,-4)--(0,0)--(4,0)); label("$4$", (2,0), S); [/asy]](http://latex.artofproblemsolving.com/3/2/6/32649d4753b5a48a0eba47330652ed5fc5a401a7.png)
Solution
Notice that when the cone is created, the radius of the circle will become the slant height of the cone and the intact circumference of the circle will become the circumference of the base of the cone.
We can calculate that the intact circumference of the circle is 
. Since that is also equal to the circumference of the cone, the radius of the cone is 
. We also have that the slant height of the cone is . Therefore, we use the Pythagorean Theorem to calculate that the height of the cone is 
. The volume of the cone is 
-PCChess
See Also
| 2020 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 9 |
Followed by Problem 11 | |
| 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 | ||
| 2020 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 8 |
Followed by Problem 10 |
| 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 12 Problems and Solutions | |
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