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Difference between revisions of "2019 AMC 8 Problems/Problem 9"

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==Problem 9==
 
==Problem 9==
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Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are <math>6</math> cm in diameter and <math>12</math> cm high. Felicia buys cat food in cylindrical cans that are <math>12</math> cm in diameter and <math>6</math> cm high. What is the ratio of the volume one of Alex's cans to the volume one of Felicia's cans?
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Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are <math>6</math> cm in diameter and <math>12</math> cm high. Felicia buys cat food in cylindrical cans that are <math>12</math> cm in diameter and <math>6</math> cm high. What is the ratio of the volume of one of Alex's cans to the volume of one of Felicia's cans?
  
 
<math>\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1</math>
 
<math>\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1</math>
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==Solution 1==
 
==Solution 1==
  
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Using the formula for the volume of a cylinder, we get Alex, <math>\pi108</math>, and Felicia, <math>\pi216</math>. We can quickly notice that <math>\pi</math> cancels out on both sides, and that Alex's volume is <math>1/2</math> of Felicia's leaving <math>1/2 = \boxed{1:2}</math> as the answer.  
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Using the formula for the volume of a cylinder, we get Alex, <math>108\pi</math>, and Felicia, <math>216\pi</math>. We can quickly notice that <math>\pi</math> cancels out on both sides and that Alex's volume is <math>1/2</math> of Felicia's leaving <math>1/2 = \boxed{1:2}</math> as the answer.  
  
 
~aopsav
 
~aopsav
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==Solution 2==
 
==Solution 2==
  
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Using the formula for the volume of a cylinder, we get that the volume of Alex's can is <math>3^2\cdot12\cdot\pi</math>, and that the volume of Felicia's can is <math>6^2\cdot6\cdot\pi</math>. Now we divide the volume of Alex's can by the volume of Felicia's can, so we get <math>\frac{1}{2}</math>, which is <math>\boxed{\textbf{(B)}\ 1:2}</math>    
+
Using the formula for the volume of a cylinder, we get that the volume of Alex's can is <math>3^2\cdot12\cdot\pi</math>, and that the volume of Felicia's can is <math>6^2\cdot6\cdot\pi</math>. Now, we divide the volume of Alex's can by the volume of Felicia's can, so we get <math>\frac{1}{2}</math>, which is <math>\boxed{\textbf{(B)}\ 1:2}</math>.   
 
              
 
              
−
lol this is something no one should be able to do.-(Algebruh123)2020
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-(Algebruh123)2020
  
 
==Solution 3==
 
==Solution 3==
  
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The ratio of the numbers is <math>1/2</math>. Looking closely at the formula <math>r^2 * h * \pi</math>, we see that the <math>r * h * \pi</math> will cancel, meaning that the ratio of them will be <math>\frac{1(2)}{2(2)}</math> = <math>\boxed{\textbf{(B)}\ 1:2}</math>  
+
The ratio of the numbers is <math>1/2</math>. Looking closely at the formula <math>r^2 * h * \pi</math>, we see that the <math>r * h * \pi</math> will cancel, meaning that the ratio of them will be <math>\frac{1(2)}{2(2)}</math> = <math>\boxed{\textbf{(B)}\ 1:2}</math>.
  
 
-Lcz
 
-Lcz
 +
 +
 +
== Solution 4 ==
 +
 +
The second can is <math>\cdot 2</math> size in each of 2 dimensions, and <math>\cdot 1/2</math> size in 1 dimension. <math> 2^2/2
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= \boxed{\textbf{(B)}\ 1:2}</math>.
 +
 +
~oinava
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 +
== Solution 5 ==
 +
 +
Without calculating much, you can do
 +
(<math>\pi ra^2) \cdot ha</math> <-- which is Alex's volume, with ra being Alex's radius<math> (1/2 \cdot</math> diameter), and <math>ha</math> being her cylinders height<math>
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(\pi rf^2) \cdot hf <-- </math>which is Felicia's volume, with <math>rf</math> being Felicia's radius, and <math>hf</math> being her cylinders height.
 +
Since we need the ratio between Alexa's and Felicias, we can do <math>(\pi ra^2)\cdot ha/(\pi rf^2)\cdot hf</math> The <math>\pi</math> cancel out, then substitute back in the numbers, which gives you:
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<math>(3^2 \cdot 12)/(6^2 \cdot 6) = (9 \cdot 12)/(36 \cdot 6) = 18/36 = 1/2 = 1:2</math>
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 +
-wahahaqueenie
  
 
== Video Solution ==
 
== Video Solution ==
 +
 +
==Video Solution by Math-X (Extremely simple approach!!!)==
 +
https://youtu.be/IgpayYB48C4?si=wsD8LhZK8hsWd9wu&t=2773
 +
 +
~Math-X
 +
 +
 
The Learning Royal : https://youtu.be/8njQzoztDGc
 
The Learning Royal : https://youtu.be/8njQzoztDGc
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== Video Solution 2 ==
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== Video Solution by OmegaLearn ==
 
https://youtu.be/FDgcLW4frg8?t=2440
 
https://youtu.be/FDgcLW4frg8?t=2440
  
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Solution detailing how to solve the problem: https://www.youtube.com/watch?v=G-gEdWP0S9M&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=10
 
Solution detailing how to solve the problem: https://www.youtube.com/watch?v=G-gEdWP0S9M&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=10
 +
 +
==Video Solution==
 +
https://youtu.be/FLT3iOKBC8c
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 +
~savannahsolver
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 +
== Video Solution ==
 +
https://youtu.be/ChwC1Hnk_pw
 +
 +
~Education, the Study of Everything
 +
 +
==Video Solution by The Power of Logic(Problem 1 to 25 Full Solution)==
 +
https://youtu.be/Xm4ZGND9WoY
 +
 +
~Hayabusa1
  
 
==See also==
 
==See also==

Latest revision as of 15:37, 30 December 2023

Problem 9

Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are $6$ cm in diameter and $12$ cm high. Felicia buys cat food in cylindrical cans that are $12$ cm in diameter and $6$ cm high. What is the ratio of the volume of one of Alex's cans to the volume of one of Felicia's cans?

$\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1$

Solution 1

Using the formula for the volume of a cylinder, we get Alex, $108\pi$, and Felicia, $216\pi$. We can quickly notice that $\pi$ cancels out on both sides and that Alex's volume is $1/2$ of Felicia's leaving $1/2 = \boxed{1:2}$ as the answer.

~aopsav

Solution 2

Using the formula for the volume of a cylinder, we get that the volume of Alex's can is $3^2\cdot12\cdot\pi$, and that the volume of Felicia's can is $6^2\cdot6\cdot\pi$. Now, we divide the volume of Alex's can by the volume of Felicia's can, so we get $\frac{1}{2}$, which is $\boxed{\textbf{(B)}\ 1:2}$.

-(Algebruh123)2020

Solution 3

The ratio of the numbers is $1/2$. Looking closely at the formula $r^2 * h * \pi$, we see that the $r * h * \pi$ will cancel, meaning that the ratio of them will be $\frac{1(2)}{2(2)}$ = $\boxed{\textbf{(B)}\ 1:2}$.

-Lcz


Solution 4

The second can is $\cdot 2$ size in each of 2 dimensions, and $\cdot 1/2$ size in 1 dimension. $2^2/2 = \boxed{\textbf{(B)}\ 1:2}$.

~oinava

Solution 5

Without calculating much, you can do ($\pi ra^2) \cdot ha$ <-- which is Alex's volume, with ra being Alex's radius$(1/2 \cdot$ diameter), and $ha$ being her cylinders height$(\pi rf^2) \cdot hf <--$which is Felicia's volume, with $rf$ being Felicia's radius, and $hf$ being her cylinders height. Since we need the ratio between Alexa's and Felicias, we can do $(\pi ra^2)\cdot ha/(\pi rf^2)\cdot hf$ The $\pi$ cancel out, then substitute back in the numbers, which gives you:

$(3^2 \cdot 12)/(6^2 \cdot 6) = (9 \cdot 12)/(36 \cdot 6) = 18/36 = 1/2 = 1:2$

-wahahaqueenie

Video Solution

Video Solution by Math-X (Extremely simple approach!!!)

https://youtu.be/IgpayYB48C4?si=wsD8LhZK8hsWd9wu&t=2773

~Math-X


The Learning Royal : https://youtu.be/8njQzoztDGc

Video Solution by OmegaLearn

https://youtu.be/FDgcLW4frg8?t=2440

~ pi_is_3.14

Video Solution

Solution detailing how to solve the problem: https://www.youtube.com/watch?v=G-gEdWP0S9M&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=10

Video Solution

https://youtu.be/FLT3iOKBC8c

~savannahsolver

Video Solution

https://youtu.be/ChwC1Hnk_pw

~Education, the Study of Everything

Video Solution by The Power of Logic(Problem 1 to 25 Full Solution)

https://youtu.be/Xm4ZGND9WoY

~Hayabusa1

See also

2019 AMC 8 (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 AJHSME/AMC 8 Problems and Solutions

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