2020 AMC 10B Problems/Problem 2: Difference between revisions

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

2020 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 1	Followed by Problem 3
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

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: Unable to save thumbnail to destination

@@ Line 14: / Line 14: @@
 ~quacker88
-==Solution 2==
+==Solution 2 (more in depth about Solution 1)==
 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.
@@ Line 25: / Line 25: @@
 ~MrThinker
+~<B+
 ==Video Solution by Education, the study of everything==

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