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 12:58, 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
