Difference between revisions of "2021 AMC 12B Problems/Problem 6"
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==Problem== | ==Problem== | ||
− | An inverted cone with base radius <math>12\mathrm{cm}</math> and height <math>18\mathrm{cm}</math> is full of water. The water is poured into a tall cylinder whose horizontal base has radius of <math>24\mathrm{cm}</math>. What is the height in centimeters of the water in the cylinder? | + | An inverted cone with base radius <math>12 \mathrm{cm}</math> and height <math>18 \mathrm{cm}</math> is full of water. The water is poured into a tall cylinder whose horizontal base has radius of <math>24 \mathrm{cm}</math>. What is the height in centimeters of the water in the cylinder? |
<math>\textbf{(A)} ~1.5 \qquad\textbf{(B)} ~3 \qquad\textbf{(C)} ~4 \qquad\textbf{(D)} ~4.5 \qquad\textbf{(E)} ~6</math> | <math>\textbf{(A)} ~1.5 \qquad\textbf{(B)} ~3 \qquad\textbf{(C)} ~4 \qquad\textbf{(D)} ~4.5 \qquad\textbf{(E)} ~6</math> | ||
+ | |||
==Solution== | ==Solution== | ||
The volume of a cone is <math>\frac{1}{3} \cdot\pi \cdot r^2 \cdot h</math> where <math>r</math> is the base radius and <math>h</math> is the height. The water completely fills up the cone so the volume of the water is <math>\frac{1}{3}\cdot18\cdot144\pi = 6\cdot144\pi</math>. | The volume of a cone is <math>\frac{1}{3} \cdot\pi \cdot r^2 \cdot h</math> where <math>r</math> is the base radius and <math>h</math> is the height. The water completely fills up the cone so the volume of the water is <math>\frac{1}{3}\cdot18\cdot144\pi = 6\cdot144\pi</math>. |
Revision as of 13:58, 13 February 2021
Contents
Problem
An inverted cone with base radius and height is full of water. The water is poured into a tall cylinder whose horizontal base has radius of . What is the height in centimeters of the water in the cylinder?
Solution
The volume of a cone is where is the base radius and is the height. The water completely fills up the cone so the volume of the water is .
The volume of a cylinder is so the volume of the water in the cylinder would be .
We can equate these two expressions because the water volume stays the same like this . We get and .
So the answer is
--abhinavg0627
Video Solution by Punxsutawney Phil
https://youtube.com/watch?v=qpvS2PVkI8A&t=509s
Video Solution by OmegaLearn (3D Geometry - Cones and Cylinders)
~ pi_is_3.14
Video Solution by Hawk Math
https://www.youtube.com/watch?v=VzwxbsuSQ80
See Also
2021 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 5 |
Followed by Problem 7 |
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 12 Problems and Solutions |
2021 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
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 |
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