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Difference between revisions of "2012 AMC 10B Problems/Problem 17"

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==Problem 17==
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==Problem==
 
Jesse cuts a circular paper disk of radius 12 along two radii to form two sectors, the smaller having a central angle of 120 degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller cone to that of the larger?
 
Jesse cuts a circular paper disk of radius 12 along two radii to form two sectors, the smaller having a central angle of 120 degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller cone to that of the larger?
  
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<math>\mathbf{(A)}</math> <math>\dfrac{1}{8}</math> <math>\space</math> <math>\mathbf{(B)}</math> <math>\dfrac{1}{4}</math> <math>\space</math> <math>\mathbf{(C)}</math> <math>\dfrac{\sqrt{10}}{10}</math> <math>\space</math> <math>\mathbf{(D)}</math> <math>\dfrac{\sqrt{5}}{6}</math> <math>\space</math> <math>\mathbf{(E)}</math> <math>\dfrac{\sqrt{5}}{5}</math>
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<math>\text{(A)} \frac{1}{8} \qquad \text{(B)} \frac{1}{4} \qquad \text{(C)} \frac{\sqrt{10}}{10} \qquad \text{(D)} \frac{\sqrt{5}}{6} \qquad \text{(E)} \frac{\sqrt{5}}{5}</math>
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[[Category: Introductory Geometry Problems]]
  
 
==Solution==
 
==Solution==
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Let's find the volume of the smaller cone first. We know that the circumference of the paper disk is <math>24\pi</math>, so the circumference of the smaller cone would be <math>\dfrac{120}{360} \times 24\pi = 8\pi</math>. This means that the radius of the smaller cone is <math>4</math>. Since the radius of the paper disk is <math>12</math>, the slant height if the smaller cone would be <math>12</math>. By the Pythagorean Theorem, the height of the cone is <math>\sqrt{12^2-4^2}=8\sqrt{2}</math>. Thus, the volume of the smaller cone is <math>\dfrac{1}{3} \times 4^2\pi \times 8\sqrt{2}=\dfrac{128\sqrt{2}}{3}\pi</math>.
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Let's find the volume of the smaller cone first. We know that the circumference of the paper disk is <math>24\pi</math>, so the circumference of the smaller cone would be <math>\dfrac{120}{360} \times 24\pi = 8\pi</math>. This means that the radius of the smaller cone is <math>4</math>. Since the radius of the paper disk is <math>12</math>, the slant height of the smaller cone would be <math>12</math>. By the Pythagorean Theorem, the height of the cone is <math>\sqrt{12^2-4^2}=8\sqrt{2}</math>. Thus, the volume of the smaller cone is <math>\dfrac{1}{3} \times 4^2\pi \times 8\sqrt{2}=\dfrac{128\sqrt{2}}{3}\pi</math>.
  
 
Now, we need to find the volume of the larger cone. Using the same reason as above, we get that the radius is <math>8</math> and the slant height is <math>12</math>. By the Pythagorean Theorem again, the height is <math>\sqrt{12^2-8^2}=4\sqrt{5}</math>. Thus, the volume of the larger cone is <math>\dfrac{1}{3} \times 8^2\pi \times 4\sqrt{5} = \dfrac{256\sqrt{5}}{3}\pi</math>.
 
Now, we need to find the volume of the larger cone. Using the same reason as above, we get that the radius is <math>8</math> and the slant height is <math>12</math>. By the Pythagorean Theorem again, the height is <math>\sqrt{12^2-8^2}=4\sqrt{5}</math>. Thus, the volume of the larger cone is <math>\dfrac{1}{3} \times 8^2\pi \times 4\sqrt{5} = \dfrac{256\sqrt{5}}{3}\pi</math>.
  
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The question asked for the ratio of the volume of the smaller cone to the larger cone. We need to find <math>\dfrac{\dfrac{128\sqrt{2}}{3}\pi}{\dfrac{256\sqrt{5}}{3}\pi}=\dfrac{\sqrt{10}}{10}</math> after simplifying, or <math>\boxed{C}</math> .
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The question asked for the ratio of the volume of the smaller cone to the larger cone. We need to find <math>\dfrac{\dfrac{128\sqrt{2}}{3}\pi}{\dfrac{256\sqrt{5}}{3}\pi}=\dfrac{\sqrt{10}}{10}</math> after simplifying, or <math>\boxed{(C)  \frac{\sqrt{10}}{10}}</math> .
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*A side note
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We can first simplify the volume ratio: <math>\frac{V_1}{V_2} = \frac{(r_1)^2 \cdot h_1}{(r_2)^2 \cdot h_2}.</math> Now we can find the GENERAL formulas for <math>r</math> and <math>h</math> based on the original circle radius and angle to cut out, then we can substitute the appropriate numbers, which gives us <math>C.</math>
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==See Also==
 
==See Also==
  
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{{AMC10 box|year=2012|ab=B|num-b=15|num-a=17}}
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{{AMC10 box|year=2012|ab=B|num-b=16|num-a=18}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 23:51, 16 January 2021

Problem

Jesse cuts a circular paper disk of radius 12 along two radii to form two sectors, the smaller having a central angle of 120 degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller cone to that of the larger?

$\text{(A)} \frac{1}{8} \qquad \text{(B)} \frac{1}{4} \qquad \text{(C)} \frac{\sqrt{10}}{10} \qquad \text{(D)} \frac{\sqrt{5}}{6} \qquad \text{(E)} \frac{\sqrt{5}}{5}$

Solution

Let's find the volume of the smaller cone first. We know that the circumference of the paper disk is $24\pi$, so the circumference of the smaller cone would be $\dfrac{120}{360} \times 24\pi = 8\pi$. This means that the radius of the smaller cone is $4$. Since the radius of the paper disk is $12$, the slant height of the smaller cone would be $12$. By the Pythagorean Theorem, the height of the cone is $\sqrt{12^2-4^2}=8\sqrt{2}$. Thus, the volume of the smaller cone is $\dfrac{1}{3} \times 4^2\pi \times 8\sqrt{2}=\dfrac{128\sqrt{2}}{3}\pi$.

Now, we need to find the volume of the larger cone. Using the same reason as above, we get that the radius is $8$ and the slant height is $12$. By the Pythagorean Theorem again, the height is $\sqrt{12^2-8^2}=4\sqrt{5}$. Thus, the volume of the larger cone is $\dfrac{1}{3} \times 8^2\pi \times 4\sqrt{5} = \dfrac{256\sqrt{5}}{3}\pi$.

The question asked for the ratio of the volume of the smaller cone to the larger cone. We need to find $\dfrac{\dfrac{128\sqrt{2}}{3}\pi}{\dfrac{256\sqrt{5}}{3}\pi}=\dfrac{\sqrt{10}}{10}$ after simplifying, or $\boxed{(C) \frac{\sqrt{10}}{10}}$ .

  • A side note

We can first simplify the volume ratio: $\frac{V_1}{V_2} = \frac{(r_1)^2 \cdot h_1}{(r_2)^2 \cdot h_2}.$ Now we can find the GENERAL formulas for $r$ and $h$ based on the original circle radius and angle to cut out, then we can substitute the appropriate numbers, which gives us $C.$

See Also

2012 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
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All AMC 10 Problems and Solutions

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