2021 AMC 12A Problems/Problem 10
Contents
Problem
Two right circular cones with vertices facing down as shown in the figure below contains the same amount of liquid. The radii of the tops of the liquid surfaces are cm and
cm. Into each cone is dropped a spherical marble of radius
cm, which sinks to the bottom and is completely submerged without spilling any liquid. What is the ratio of the rise of the liquid level in the narrow cone to the rise of the liquid level in the wide cone?
Solution
I will be referring to the areas filled with liquid as the cones.
The area of a cone is , where
is the radius of the cone and
is the height. Since the first cone has half the radius of the second cone, and both cones have the same volume, the ratio of the height of the first cone to the height of the second cone is
. After marbles are dropped, the volumes are still equal, so the ratio of the heights is still
. Therefore, the ratio of the liquid rise is
.
Solution 2 (Fraction Trick)
Initially:
For the narrow cone liquid, the base radius is Let its height be
By similar triangles, the ratio of base radius to height is
The volume is
For the wide cone liquid, the base radius is Let its height be
By similar triangles, the ratio of base radius to height is
The volume is
Equating initial volumes gives from which
Finally:
For the narrow cone liquid, the base radius is where
By similar triangles, it follows that its height is
and its volume is
For the wide cone liquid, the base radius is where
By similar triangles, it follows that its height is
and its volume is
Equating final volumes simplifies to or
Lastly, the fraction we seek is
PS:
1. This problem uses the following fraction trick:
For unequal positive numbers and
if
then
Quick Proof
From we know that
and
. Therefore,
2. Most of the steps can be done through mental math. Also, drawing a table showing the initial and final measurements can effectively organize the work.
~MRENTHUSIASM
Video Solution by Hawk Math
https://www.youtube.com/watch?v=AjQARBvdZ20
Video Solution by OmegaLearn (Similar Triangles, 3D Geometry - Cones)
~ pi_is_3.14
See also
2021 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 9 |
Followed by Problem 11 |
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All AMC 12 Problems and Solutions |
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