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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>.
  
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==Solution 2==
 
==Solution 2==
  
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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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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 5 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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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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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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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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Lastly, to find the total of Carl's and Kate's cubes, you must add the total volume of these people. This is going to be <math>5+40=\boxed{\textbf{(E)} ~45}</math>.
  
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==Video Solution==
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- BrightPorcupine
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~MrThinker
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 +
==Video Solution (HOW TO THINK CRITICALLY!!!)==
 +
https://www.youtube.com/watch?v=ExEfaIOqt_w
  
 
Check It Out! :)
 
Check It Out! :)
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Education, the study of everything  
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~Education, the study of everything  
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https://www.youtube.com/watch?v=ExEfaIOqt_w
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https://youtu.be/Gkm5rU5MlOU
 
https://youtu.be/Gkm5rU5MlOU
  
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~IceMatrix
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https://youtu.be/FcPO4EXDwzc
 
https://youtu.be/FcPO4EXDwzc
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https://www.youtube.com/watch?v=GNPAgQ8fSP0&t=1s
 
https://www.youtube.com/watch?v=GNPAgQ8fSP0&t=1s
  
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~alexexplains
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~AlexExplains
  
 
==See Also==
 
==See Also==

Latest revision as of 13:32, 6 June 2023

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

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 these people. This is going to be $5+40=\boxed{\textbf{(E)} ~45}$.

- BrightPorcupine

~MrThinker

Video Solution (HOW TO THINK CRITICALLY!!!)

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

Check It Out! :) ~Education, the study of everything


https://youtu.be/Gkm5rU5MlOU


https://youtu.be/FcPO4EXDwzc

~savannahsolver

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

~AlexExplains

See Also

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

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