Difference between revisions of "2021 AMC 12A Problems/Problem 10"
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==Problem== | ==Problem== | ||
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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 <math>3</math> cm and <math>6</math> cm. Into each cone is dropped a spherical marble of radius <math>1</math> 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? | 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 <math>3</math> cm and <math>6</math> cm. Into each cone is dropped a spherical marble of radius <math>1</math> 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? | ||
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</asy> | </asy> | ||
| − | <math>\textbf{(A) } | + | <math>\textbf{(A) }1 \qquad \textbf{(B) }\frac{47}{43 \qquad \textbf{(C) }2 \qquad \textbf{(D) }\frac{40}{13 \qquad \textbf{(E) }4</math> |
==Solution== | ==Solution== | ||
| − | The answer is <math>\boxed{\textbf{(E) } 4 | + | The answer is <math>\boxed{\textbf{(E) } 4}</math> |
==See also== | ==See also== | ||
{{AMC12 box|year=2021|ab=A|num-b=9|num-a=11}} | {{AMC12 box|year=2021|ab=A|num-b=9|num-a=11}} | ||
| + | |||
| + | [[Category:Introductory Geometry Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 16:39, 11 February 2021
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?
![[asy] size(350); defaultpen(linewidth(0.8)); real h1 = 10, r = 3.1, s=0.75; pair P = (r,h1), Q = (-r,h1), Pp = s * P, Qp = s * Q; path e = ellipse((0,h1),r,0.9), ep = ellipse((0,h1*s),r*s,0.9); draw(ellipse(origin,r*(s-0.1),0.8)); fill(ep,gray(0.8)); fill(origin--Pp--Qp--cycle,gray(0.8)); draw((-r,h1)--(0,0)--(r,h1)^^e); draw(subpath(ep,0,reltime(ep,0.5)),linetype("4 4")); draw(subpath(ep,reltime(ep,0.5),reltime(ep,1))); draw(Qp--(0,Qp.y),Arrows(size=8)); draw(origin--(0,12),linetype("4 4")); draw(origin--(r*(s-0.1),0)); label("$3$",(-0.9,h1*s),N,fontsize(10)); real h2 = 7.5, r = 6, s=0.6, d = 14; pair P = (d+r-0.05,h2-0.15), Q = (d-r+0.05,h2-0.15), Pp = s * P + (1-s)*(d,0), Qp = s * Q + (1-s)*(d,0); path e = ellipse((d,h2),r,1), ep = ellipse((d,h2*s+0.09),r*s,1); draw(ellipse((d,0),r*(s-0.1),0.8)); fill(ep,gray(0.8)); fill((d,0)--Pp--Qp--cycle,gray(0.8)); draw(P--(d,0)--Q^^e); draw(subpath(ep,0,reltime(ep,0.5)),linetype("4 4")); draw(subpath(ep,reltime(ep,0.5),reltime(ep,1))); draw(Qp--(d,Qp.y),Arrows(size=8)); draw((d,0)--(d,10),linetype("4 4")); draw((d,0)--(d+r*(s-0.1),0)); label("$6$",(d-r/4,h2*s-0.06),N,fontsize(10)); [/asy]](http://latex.artofproblemsolving.com/8/1/d/81d626eb47050e0f212495e407860357b3a98a6c.png)
$\textbf{(A) }1 \qquad \textbf{(B) }\frac{47}{43 \qquad \textbf{(C) }2 \qquad \textbf{(D) }\frac{40}{13 \qquad \textbf{(E) }4$ (Error compiling LaTeX. Unknown error_msg)
Solution
The answer is 
See also
| 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. 
