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Difference between revisions of "2020 AMC 10B Problems/Problem 2"

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<math>\textbf{(A)}\ 24 \qquad\textbf{(B)}\ 25 \qquad\textbf{(C)}\ 28 \qquad\textbf{(D)}\ 40 \qquad\textbf{(E)}\ 45</math>
 
<math>\textbf{(A)}\ 24 \qquad\textbf{(B)}\ 25 \qquad\textbf{(C)}\ 28 \qquad\textbf{(D)}\ 40 \qquad\textbf{(E)}\ 45</math>
  
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==Solution==
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==Solution 1==
 
A cube with side length <math>1</math> has volume <math>1^3=1</math>, so <math>5</math> of these will have a total volume of <math>5\cdot1=5</math>.
 
A cube with side length <math>1</math> has volume <math>1^3=1</math>, so <math>5</math> of these will have a total volume of <math>5\cdot1=5</math>.
  
 
A cube with side length <math>2</math> has volume <math>2^3=8</math>, so <math>5</math> of these will have a total volume of <math>5\cdot8=40</math>.
 
A cube with side length <math>2</math> has volume <math>2^3=8</math>, so <math>5</math> of these will have a total volume of <math>5\cdot8=40</math>.
  
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<math>5+40=\boxed{\textbf{(E) }45}</math> ~quacker88
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<math>5+40=\boxed{\textbf{(E) }45}</math>.
  
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==Solution 2==
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~quacker88
  
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The total volume of Carl's cubes is 5. This is because to find the volume of a cube or a rectangular prism, you have to multiply the height by the length by the width. So in this question, it would be 1*1*1. This is equal to 1. Since Carl has 5 cubes, you will have to multiply 1 by 5, to account for all the 5 cubes.
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==Solution 2 (more in depth about Solution 1)==
  
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Next, to find the total volume of Kate's cubes you have to do the same thing. Except, this time, the height, the width, and the length, are all 2, so it will be 2^3. This is equal to 8. Now you have to multiply by 5 to account for all the 5 blocks. This is 40. So the total volume of Kate's cubes are 40.
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The total volume of Carl's cubes is <math>5</math>. This is because to find the volume of a cube or a rectangular prism, you have to multiply the height by the length by the width. So in this question, it would be <math>1 \times 1 \times 1</math>. This is equal to <math>1</math>. Since Carl has <math>5</math> cubes, you will have to multiply <math>1</math> by <math>5</math>, to account for all the <math>5</math> cubes.
  
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Lastly, to find the total of both Carl's and Kate's cubes, all you have to do it add the total volume of these people. This is going to be 5+40. And this is 45. - BrightPorcupine
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Next, to find the total volume of Kate's cubes you have to do the same thing. Except, this time, the height, the width, and the length, are all <math>2</math>, so it will be <math>2 \times 2 \times 2 = 8.</math> Now you have to multiply by <math>5</math> to account for all the <math>5</math> blocks. This is <math>40</math>. So the total volume of Kate's cubes is <math>40</math>.
  
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==Video Solution==
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Lastly, to find the total of Carl's and Kate's cubes, you must add the total volume of their cubes together. This is going to be <math>5+40=\boxed{\textbf{(E)} ~45}</math>.
  
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Check It Out! :)
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~BrightPorcupine
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Education, the study of everything  
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~MrThinker
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~<B+
 +
 
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==Video Solution by Education, the study of everything==
 
https://www.youtube.com/watch?v=ExEfaIOqt_w
 
https://www.youtube.com/watch?v=ExEfaIOqt_w
  
 +
~Education, the study of everything
  
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==Video Solution by TheBeautyofMath==
 
https://youtu.be/Gkm5rU5MlOU
 
https://youtu.be/Gkm5rU5MlOU
  
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~IceMatrix
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==Video Solution by WhyMath==
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https://youtu.be/FcPO4EXDwzc
 
https://youtu.be/FcPO4EXDwzc
  
 
~savannahsolver
 
~savannahsolver
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==Video Solution by AlexExplains==
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https://www.youtube.com/watch?v=GNPAgQ8fSP0&t=1s
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~AlexExplains
  
 
==See Also==
 
==See Also==

Latest revision as of 07:26, 15 August 2024

Problem

Carl has $5$ cubes each having side length $1$, and Kate has $5$ cubes each having side length $2$. What is the total volume of these $10$ cubes?

$\textbf{(A)}\ 24 \qquad\textbf{(B)}\ 25 \qquad\textbf{(C)}\ 28 \qquad\textbf{(D)}\ 40 \qquad\textbf{(E)}\ 45$

Solution 1

A cube with side length $1$ has volume $1^3=1$, so $5$ of these will have a total volume of $5\cdot1=5$.

A cube with side length $2$ has volume $2^3=8$, so $5$ of these will have a total volume of $5\cdot8=40$.

$5+40=\boxed{\textbf{(E) }45}$.

~quacker88

Solution 2 (more in depth about Solution 1)

The total volume of Carl's cubes is $5$. This is because to find the volume of a cube or a rectangular prism, you have to multiply the height by the length by the width. So in this question, it would be $1 \times 1 \times 1$. This is equal to $1$. Since Carl has $5$ cubes, you will have to multiply $1$ by $5$, to account for all the $5$ cubes.

Next, to find the total volume of Kate's cubes you have to do the same thing. Except, this time, the height, the width, and the length, are all $2$, so it will be $2 \times 2 \times 2 = 8.$ Now you have to multiply by $5$ to account for all the $5$ blocks. This is $40$. So the total volume of Kate's cubes is $40$.

Lastly, to find the total of Carl's and Kate's cubes, you must add the total volume of their cubes together. This is going to be $5+40=\boxed{\textbf{(E)} ~45}$.

~BrightPorcupine

~MrThinker

~<B+

Video Solution by Education, the study of everything

https://www.youtube.com/watch?v=ExEfaIOqt_w

~Education, the study of everything

Video Solution by TheBeautyofMath

https://youtu.be/Gkm5rU5MlOU

Video Solution by WhyMath

https://youtu.be/FcPO4EXDwzc

~savannahsolver

Video Solution by AlexExplains

https://www.youtube.com/watch?v=GNPAgQ8fSP0&t=1s

~AlexExplains

See Also

2020 AMC 10B (Problems • Answer Key • Resources)
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Problem 3
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All AMC 10 Problems and Solutions

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