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Difference between revisions of "2020 AMC 10B Problems/Problem 10"

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==Problem 10==
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{{duplicate|[[2020 AMC 10B Problems|2020 AMC 10B #10]] and [[2020 AMC 12B Problems|2020 AMC 12B #9]]}}
  
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== Problem ==
 
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?
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<asy>
 
<asy>
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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));
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label("$4$", (2,0), S);
 
label("$4$", (2,0), S);
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</asy>
  
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</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>
  
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==Solution==
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== Solutions ==
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=== Solution 1 ===
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Notice that when the cone is created, the 2 shown radii when merged 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.
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We can calculate that the intact circumference of the circle is <math>8\pi\cdot\frac{3}{4}=6\pi</math>. Since that is also equal to the circumference of the cone, the radius of the cone is <math>3</math>. We also have that the slant height of the cone is <math>4</math>. Therefore, we use the Pythagorean Theorem to calculate that the height of the cone is <math>\sqrt{4^2-3^2}=\sqrt7</math>. The volume of the cone is <math>\frac{1}{3}\cdot\pi\cdot3^2\cdot\sqrt7=\boxed{\textbf{(C)}\ 3 \pi \sqrt7 }</math>
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-PCChess
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=== Solution 2 (Last Resort/Cheap) ===
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Using a ruler, measure a circle of radius 4 and cut out the circle and then the quarter missing. Then, fold it into a cone and measure the diameter to be 6 cm <math>\implies r=3</math>. You can form a right triangle with sides 3, 4, and then through the Pythagorean theorem the height <math>h</math> is found to be <math>h^2 = 4^{2} - 3^{2} \implies h = \sqrt{7}</math>. The volume of a cone is <math>\frac{1}{3}\pi r^{2}h</math>. Plugging in we find <math>V = 3\pi \sqrt{7} \implies \boxed{\textbf{(C)}}</math>
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- DBlack2021
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=== Solution 3 ===
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The radius of the given <math>\frac{3}{4}</math> - circle will end up being the slant height of the cone. Thus, the radius and height of the cone are legs of a right triangle with hypotenuse <math>4</math>. The volume of a cone is <math>\frac{1}{3}\pi r^{2}h</math>. Using this with the options, we can take out the <math>\pi</math> and multiply the coefficient of the radical by <math>3</math> to get <math>r^{2}h</math>. We can then use the <math>r</math> and <math>h</math> values to see that the only option that satisfies <math>r^2+h^2=4^2=16</math> is <math>\boxed{\textbf{(C)}}</math>
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- ColtsFan10
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==Video Solution (HOW TO THINK CREATIVELY!!!)==
 +
https://youtu.be/zQ9GHc6u6dM
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 +
~Education, the Study of Everything
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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.
 
  
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We can calculate that the intact circumference of the circle is <math>8\pi\cdot\frac{3}{4}=6\pi</math>. Since that is also equal to the circumference of the cone, the radius of the cone is <math>3</math>. We also have that the slant height of the cone is <math>4</math>. Therefore, we use the Pythagorean Theorem to calculate that the height of the cone is <math>\sqrt{4^2-3^2}=\sqrt7</math>. The volume of the cone is <math>\frac{1}{3}\cdot\pi\cdot3^2\cdot\sqrt7=\boxed{\textbf{(C)}\ 3 \pi \sqrt7 }</math> -PCChess
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== Video Solution by TheBeautyOfMath ==
 +
https://youtu.be/OHR_6U686Qg
  
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==Video Solution==
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https://youtu.be/6ujfjGLzVoE
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https://youtu.be/OHR_6U686Qg
 
  
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~IceMatrix
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== Video Solution by WhyMath ==
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https://youtu.be/4OAhfceUJXc
  
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==See Also==
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~savannahsolver
  
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== See Also ==
 
{{AMC10 box|year=2020|ab=B|num-b=9|num-a=11}}
 
{{AMC10 box|year=2020|ab=B|num-b=9|num-a=11}}
 
{{AMC12 box|year=2020|ab=B|num-b=8|num-a=10}}
 
{{AMC12 box|year=2020|ab=B|num-b=8|num-a=10}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 07:42, 15 August 2024

The following problem is from both the 2020 AMC 10B #10 and 2020 AMC 12B #9, so both problems redirect to this page.

Problem

A three-quarter sector of a circle of radius $4$ 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]

$\textbf{(A)}\ 3\pi \sqrt5 \qquad\textbf{(B)}\ 4\pi \sqrt3 \qquad\textbf{(C)}\ 3 \pi \sqrt7 \qquad\textbf{(D)}\ 6\pi \sqrt3 \qquad\textbf{(E)}\ 6\pi \sqrt7$

Solutions

Solution 1

Notice that when the cone is created, the 2 shown radii when merged 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 $8\pi\cdot\frac{3}{4}=6\pi$. Since that is also equal to the circumference of the cone, the radius of the cone is $3$. We also have that the slant height of the cone is $4$. Therefore, we use the Pythagorean Theorem to calculate that the height of the cone is $\sqrt{4^2-3^2}=\sqrt7$. The volume of the cone is $\frac{1}{3}\cdot\pi\cdot3^2\cdot\sqrt7=\boxed{\textbf{(C)}\ 3 \pi \sqrt7 }$

-PCChess

Solution 2 (Last Resort/Cheap)

Using a ruler, measure a circle of radius 4 and cut out the circle and then the quarter missing. Then, fold it into a cone and measure the diameter to be 6 cm $\implies r=3$. You can form a right triangle with sides 3, 4, and then through the Pythagorean theorem the height $h$ is found to be $h^2 = 4^{2} - 3^{2} \implies h = \sqrt{7}$. The volume of a cone is $\frac{1}{3}\pi r^{2}h$. Plugging in we find $V = 3\pi \sqrt{7} \implies \boxed{\textbf{(C)}}$

- DBlack2021

Solution 3

The radius of the given $\frac{3}{4}$ - circle will end up being the slant height of the cone. Thus, the radius and height of the cone are legs of a right triangle with hypotenuse $4$. The volume of a cone is $\frac{1}{3}\pi r^{2}h$. Using this with the options, we can take out the $\pi$ and multiply the coefficient of the radical by $3$ to get $r^{2}h$. We can then use the $r$ and $h$ values to see that the only option that satisfies $r^2+h^2=4^2=16$ is $\boxed{\textbf{(C)}}$

- ColtsFan10

Video Solution (HOW TO THINK CREATIVELY!!!)

https://youtu.be/zQ9GHc6u6dM

~Education, the Study of Everything


Video Solution by TheBeautyOfMath

https://youtu.be/OHR_6U686Qg

https://youtu.be/6ujfjGLzVoE

Video Solution by WhyMath

https://youtu.be/4OAhfceUJXc

~savannahsolver

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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