2021 AMC 12A Problems/Problem 10
- The following problem is from both the 2021 AMC 10A #12 and 2021 AMC 12A #10, so both problems redirect to this page.
Contents
- 1 Problem
- 2 Solution 1 (Algebra)
- 3 Solution 2 (Approximate Cones with Cylinders)
- 4 Solution 3 (Calculus)
- 5 Solution 4 (Extremely Quick Observation)
- 6 Video Solution (Simple and Quick)
- 7 Video Solution by Aaron He (Algebra)
- 8 Video Solution by OmegaLearn (Similar Triangles, 3D Geometry - Cones)
- 9 Video Solution by TheBeautyofMath
- 10 Video Solution by WhyMath
- 11 See also
Problem
Two right circular cones with vertices facing down as shown in the figure below contain 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 1 (Algebra)
Initial Scenario
Let the heights of the narrow cone and the wide cone be and respectively. We have the following table: Equating the volumes gives which simplifies to
Furthermore, by similar triangles:
- For the narrow cone, the ratio of the base radius to the height is which always remains constant.
- For the wide cone, the ratio of the base radius to the height is which always remains constant.
Two solutions follow from here:
Solution 1.1 (Properties of Fractions)
Final Scenario
For the narrow cone and the wide cone, let their base radii be and (for some ), respectively. By the similar triangles discussed above, their heights must be and respectively. We have the following table: Recall that Equating the volumes gives which simplifies to or
Finally, the requested ratio is Remarks
- This solution uses the following property of fractions:
For unequal positive numbers and if then
- This solution shows that, regardless of the shape or the volume of the solid dropped into each cone, the requested ratio stays the same as long as the solid sinks to the bottom and is completely submerged without spilling any liquid.
~MRENTHUSIASM
Solution 1.2 (Bash)
Final Scenario
For the narrow cone and the wide cone, let their base radii be and respectively; let their rises of the liquid levels be and respectively. We have the following table: By the similar triangles discussed above, we get The volume of the marble dropped into each cone is
Now, we set up an equation for the volume of the narrow cone, then express in terms of Next, we set up an equation for the volume of the wide cone, then express in terms of Using a similar process from above, we get Recall that Therefore, the requested ratio is ~MRENTHUSIASM
Solution 2 (Approximate Cones with Cylinders)
The heights of the cones are not given, so suppose the heights are very large (i.e. tending towards infinity) in order to approximate the cones as cylinders with base radii and and infinitely large height. Then the base area of the wide cylinder is times that of the narrow cylinder. Since we are dropping a ball of the same volume into each cylinder, the water level in the narrow cone/cylinder should rise times as much.
~scrabbler94
Solution 3 (Calculus)
The volume of the shorter cone can be expressed as , and the volume of the larger cone can be expressed as . We also know that the volume changes by , because the volume of the cm sphere is .
Taking the derivative of the equations, we get: and .
Therefore, and . The ratio is giving us the answer of .
~aurellia
Solution 4 (Extremely Quick Observation)
Note that as long as the volumes are equal, the ratio of the heights of Cone 2 to Cone 1 is always . We can prove this as follows:
If we fix Cone 2 to have a certain height, then the volume of the cone is . Now if Cone 1 has the same volume, its height would be .
Video Solution (Simple and Quick)
~ Education, the Study of Everything
Video Solution by Aaron He (Algebra)
https://www.youtube.com/watch?v=xTGDKBthWsw&t=10m20s
Video Solution by OmegaLearn (Similar Triangles, 3D Geometry - Cones)
~ pi_is_3.14
Video Solution by TheBeautyofMath
First-this is not the most efficient solution. I did not perceive the shortcut before filming though I suspected it.
https://youtu.be/t-EEP2V4nAE?t=231 (for AMC 10A)
https://youtu.be/cckGBU2x1zg?t=814 (for AMC 12A)
~IceMatrix
Video Solution by WhyMath
~savannahsolver
See also
2021 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
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 |
2021 AMC 12A (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 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.