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Difference between revisions of "2021 AMC 12A Problems/Problem 10"

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{{duplicate|[[2021 AMC 10A Problems#Problem 12|2021 AMC 10A #12]] and [[2021 AMC 12A Problems#Problem 10|2021 AMC 12A #10]]}}
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{{duplicate|[[2021 AMC 10A Problems/Problem 12|2021 AMC 10A #12]] and [[2021 AMC 12A Problems/Problem 10|2021 AMC 12A #10]]}}
  
 
==Problem==
 
==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 <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 contain 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?
  
 
<asy>
 
<asy>
Line 63: Line 63:
 
<cmath>\begin{array}{cccccc}
 
<cmath>\begin{array}{cccccc}
 
& \textbf{Base Radius} & \textbf{Height} & & \textbf{Volume} & \\ [2ex]
 
& \textbf{Base Radius} & \textbf{Height} & & \textbf{Volume} & \\ [2ex]
−
\textbf{Narrow Cone} & 3x & h_1x & & \frac13\pi(3x)^2\left(h_1x\right)=3\pi h_1 x^3 & \\ [2ex]
+
\textbf{Narrow Cone} & 3x & h_1x & & \frac13\pi(3x)^2(h_1x)=3\pi h_1 x^3 & \\ [2ex]
−
\textbf{Wide Cone} & 6y & h_2y & & \hspace{2.25mm}\frac13\pi(6y)^2\left(h_2y\right)=12\pi h_2 y^3 &
+
\textbf{Wide Cone} & 6y & h_2y & & \hspace{2.0625mm}\frac13\pi(6y)^2(h_2y)=12\pi h_2 y^3 &
 
\end{array}</cmath>
 
\end{array}</cmath>
 
Recall that <math>\frac{h_1}{h_2}=4.</math> Equating the volumes gives <math>3\pi h_1 x^3=12\pi h_2 y^3,</math> which simplifies to <math>x^3=y^3,</math> or <math>x=y.</math>
 
Recall that <math>\frac{h_1}{h_2}=4.</math> Equating the volumes gives <math>3\pi h_1 x^3=12\pi h_2 y^3,</math> which simplifies to <math>x^3=y^3,</math> or <math>x=y.</math>
  
−
Lastly, the requested ratio is <cmath>\frac{h_1 x - h_1}{h_2 y - h_2}=\frac{h_1 (x-1)}{h_2 (y-1)}=\frac{h_1}{h_2}=\boxed{\textbf{(E) }4:1}.</cmath>
+
Finally, the requested ratio is <cmath>\frac{h_1 x - h_1}{h_2 y - h_2}=\frac{h_1 (x-1)}{h_2 (y-1)}=\frac{h_1}{h_2}=\boxed{\textbf{(E) }4:1}.</cmath>
−
 
 
 
<u><b>Remarks</b></u>
 
<u><b>Remarks</b></u>
−
 
 
<ol style="margin-left: 1.5em;">
 
<ol style="margin-left: 1.5em;">
 
   <li>This solution uses the following property of fractions: <p>  
 
   <li>This solution uses the following property of fractions: <p>  
−
For unequal positive numbers <math>a,b,c</math> and <math>d,</math> if <math>\frac ab = \frac cd = k,</math> then <math>\frac{a\pm c}{b\pm d}=k.</math> <p>
+
For unequal positive numbers <math>a,b,c</math> and <math>d,</math> if <math>\frac ab = \frac cd = k,</math> then <math>\frac{a\pm c}{b\pm d}=\frac{bk\pm dk}{b\pm d}=\frac{(b\pm d)k}{b\pm d}=k.</math></li><p>
−
We can prove this property quickly: <p>From <math>\frac ab = \frac cd = k,</math> we know that <math>a=bk</math> and <math>c=dk</math>. Therefore, we conclude that <cmath>\frac{a\pm c}{b\pm d}=\frac{bk\pm dk}{b\pm d}=\frac{\left(b\pm d\right)k}{b\pm d}=k.</cmath></li><p>
 
 
   <li>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.</li><p>
 
   <li>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.</li><p>
 
</ol>
 
</ol>
−
 
 
~MRENTHUSIASM
 
~MRENTHUSIASM
  
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\textbf{Wide Cone} & r_2 & h_2+\Delta h_2 & & \frac13\pi r_2^2(h_2+\Delta h_2) &
 
\textbf{Wide Cone} & r_2 & h_2+\Delta h_2 & & \frac13\pi r_2^2(h_2+\Delta h_2) &
 
\end{array}</cmath>
 
\end{array}</cmath>
−
 
 
By the similar triangles discussed above, we get
 
By the similar triangles discussed above, we get
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
Line 101: Line 96:
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
 
\frac13\pi r_1^2(h_1+\Delta h_1) &= 3\pi h_1+\frac43\pi \\
 
\frac13\pi r_1^2(h_1+\Delta h_1) &= 3\pi h_1+\frac43\pi \\
−
\frac13\pi{\biggl(\phantom{ }\underbrace{\frac{3}{h_1}(h_1+\Delta h_1)}_{\text{by }(1)}\phantom{ }\biggr)}^2(h_1+\Delta h_1) &= 3\pi h_1+\frac43\pi \\
+
\frac13 r_1^2(h_1+\Delta h_1) &= 3h_1+\frac43 \\
 +
\frac13\left(\frac{3}{h_1}(h_1+\Delta h_1)\right)^2(h_1+\Delta h_1) &= 3h_1+\frac43 &&\text{by }(1) \\
 
\frac{3}{h_1^2}(h_1+\Delta h_1)^3 &= 3h_1+\frac43 \\
 
\frac{3}{h_1^2}(h_1+\Delta h_1)^3 &= 3h_1+\frac43 \\
 
(h_1+\Delta h_1)^3 &= h_1^3 + \frac{4h_1^2}{9} \\
 
(h_1+\Delta h_1)^3 &= h_1^3 + \frac{4h_1^2}{9} \\
 
\Delta h_1 &= \sqrt[3]{h_1^3 + \frac{4h_1^2}{9}}-h_1.
 
\Delta h_1 &= \sqrt[3]{h_1^3 + \frac{4h_1^2}{9}}-h_1.
 
\end{align*}</cmath>
 
\end{align*}</cmath>
−
 
 
Next, we set up an equation for the volume of the wide cone, then express <math>\Delta h_2</math> in terms of <math>h_2:</math>
 
Next, we set up an equation for the volume of the wide cone, then express <math>\Delta h_2</math> in terms of <math>h_2:</math>
 
<cmath>\frac13\pi r_2^2(h_2+\Delta h_2) = 12\pi h_2+\frac43\pi.</cmath>
 
<cmath>\frac13\pi r_2^2(h_2+\Delta h_2) = 12\pi h_2+\frac43\pi.</cmath>
−
Using the exact same process from above (but with different numbers), we get <cmath>\Delta h_2 = \sqrt[3]{h_2^3+\frac{h_2^2}{9}}-h_2.</cmath>
+
Using a similar process from above, we get <cmath>\Delta h_2 = \sqrt[3]{h_2^3+\frac{h_2^2}{9}}-h_2.</cmath>
 
Recall that <math>\frac{h_1}{h_2}=4.</math> Therefore, the requested ratio is  
 
Recall that <math>\frac{h_1}{h_2}=4.</math> Therefore, the requested ratio is  
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
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&=\boxed{\textbf{(E) }4:1}.
 
&=\boxed{\textbf{(E) }4:1}.
 
\end{align*}</cmath>
 
\end{align*}</cmath>
−
 
 
~MRENTHUSIASM
 
~MRENTHUSIASM
  
−
==Solution 2 (Quick and Dirty)==
+
==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 3 and 6 and infinitely large height. Then the base area of the wide cylinder is 4 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 <math>\boxed{\textbf{(E) } 4}</math> times as much.
+
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 <math>3</math> and <math>6</math> and infinitely large height. Then the base area of the wide cylinder is <math>4</math> 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 <math>\boxed{\textbf{(E) } 4}</math> times as much.
 
   
 
   
−
-scrabbler94
+
~scrabbler94
 +
 
 +
==Solution 3 (Calculus)==
 +
The volume of the shorter cone can be expressed as <math>V_1=3\pi h_1</math>, and the volume of the larger cone can be expressed as <math>V_2=12\pi h_2</math>. We also know that the volume changes by <math>\frac{4\pi}{3}</math>, because the volume of the <math>1</math>cm sphere is <math>\frac{4\pi}{3}</math>.
 +
 
 +
Taking the derivative of the equations, we get: <math>dV_1=3\pi(dh_1)=\frac{4\pi}{3}</math> and <math>dV_2=12\pi(dh_2)=\frac{4\pi}{3}</math>.
 +
 
 +
Therefore, <math>dh_1=\frac{4\pi}{3}\cdot\frac{1}{3\pi}=\frac{4}{9}</math> and <math>dh_2=\frac{4\pi}{3}\cdot\frac{1}{12\pi}=\frac{1}{9}</math>. The ratio is <math>\frac{4}{9}:\frac{1}{9}</math> giving us the answer of <math>\boxed{\textbf{(E) }4:1}</math>.
 +
 
 +
~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 <math>1:4</math>. We can prove this as follows:
  
−
==Solution 3==
+
If we fix Cone 2 to have a certain height, then the volume of the cone is <math>\frac{1}{3}\cdot36\pi{h_1}=12\pi{h_1}</math>. Now if Cone 1 has the same volume, its height would be <math>\frac{12\pi{h_1}}{\frac{1}{3}\cdot9\pi}=\frac{12\pi{h_1}}{3\pi}=4h_1</math>.
−
Since the radius of the narrow cone is 1/2 the radius of the wider cone, the ratio of their areas is <math>\dfrac{1}{4}</math>. Therefore, the ratio of the height of the narrow cone to the height of the wide cone must be <math>\dfrac{4}{1}</math>. Note that this ratio is constant, regardless of how much water is dropped as long as it is an equal amount for both cones. See Solution 2 for another explanation.
 
  
 
==Video Solution (Simple and Quick)==
 
==Video Solution (Simple and Quick)==

Latest revision as of 14:09, 1 August 2023

The following problem is from both the 2021 AMC 10A #12 and 2021 AMC 12A #10, so both problems redirect to this page.

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 $3$ cm and $6$ cm. Into each cone is dropped a spherical marble of radius $1$ 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]

$\textbf{(A) }1:1 \qquad \textbf{(B) }47:43 \qquad \textbf{(C) }2:1 \qquad \textbf{(D) }40:13 \qquad \textbf{(E) }4:1$

Solution 1 (Algebra)

Initial Scenario

Let the heights of the narrow cone and the wide cone be $h_1$ and $h_2,$ respectively. We have the following table: \[\begin{array}{cccccc} & \textbf{Base Radius} & \textbf{Height} & & \textbf{Volume} & \\ [2ex] \textbf{Narrow Cone} & 3 & h_1 & & \frac13\pi(3)^2h_1=3\pi h_1 & \\ [2ex] \textbf{Wide Cone} & 6 & h_2 & & \hspace{2mm}\frac13\pi(6)^2h_2=12\pi h_2 & \end{array}\] Equating the volumes gives $3\pi h_1=12\pi h_2,$ which simplifies to $\frac{h_1}{h_2}=4.$

Furthermore, by similar triangles:

  • For the narrow cone, the ratio of the base radius to the height is $\frac{3}{h_1},$ which always remains constant.
  • For the wide cone, the ratio of the base radius to the height is $\frac{6}{h_2},$ 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 $3x$ and $6y$ (for some $x,y>1$), respectively. By the similar triangles discussed above, their heights must be $h_1x$ and $h_2y,$ respectively. We have the following table: \[\begin{array}{cccccc} & \textbf{Base Radius} & \textbf{Height} & & \textbf{Volume} & \\ [2ex] \textbf{Narrow Cone} & 3x & h_1x & & \frac13\pi(3x)^2(h_1x)=3\pi h_1 x^3 & \\ [2ex] \textbf{Wide Cone} & 6y & h_2y & & \hspace{2.0625mm}\frac13\pi(6y)^2(h_2y)=12\pi h_2 y^3 & \end{array}\] Recall that $\frac{h_1}{h_2}=4.$ Equating the volumes gives $3\pi h_1 x^3=12\pi h_2 y^3,$ which simplifies to $x^3=y^3,$ or $x=y.$

Finally, the requested ratio is \[\frac{h_1 x - h_1}{h_2 y - h_2}=\frac{h_1 (x-1)}{h_2 (y-1)}=\frac{h_1}{h_2}=\boxed{\textbf{(E) }4:1}.\] Remarks

  1. This solution uses the following property of fractions:

    For unequal positive numbers $a,b,c$ and $d,$ if $\frac ab = \frac cd = k,$ then $\frac{a\pm c}{b\pm d}=\frac{bk\pm dk}{b\pm d}=\frac{(b\pm d)k}{b\pm d}=k.$

  2. 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 $r_1$ and $r_2,$ respectively; let their rises of the liquid levels be $\Delta h_1$ and $\Delta h_2,$ respectively. We have the following table: \[\begin{array}{cccccc} & \textbf{Base Radius} & \textbf{Height} & & \textbf{Volume} & \\ [2ex] \textbf{Narrow Cone} & r_1 & h_1+\Delta h_1 & & \frac13\pi r_1^2(h_1+\Delta h_1) & \\ [2ex] \textbf{Wide Cone} & r_2 & h_2+\Delta h_2 & & \frac13\pi r_2^2(h_2+\Delta h_2) & \end{array}\] By the similar triangles discussed above, we get \begin{align*} \frac{3}{h_1}&=\frac{r_1}{h_1+\Delta h_1} &\implies \quad r_1&=\frac{3}{h_1}(h_1+\Delta h_1), & \hspace{10mm} (1) \\ \frac{6}{h_2}&=\frac{r_2}{h_2+\Delta h_2} &\implies \quad r_2&=\frac{6}{h_2}(h_2+\Delta h_2). & (2) \end{align*} The volume of the marble dropped into each cone is $\frac43\pi(1)^3=\frac43\pi.$

Now, we set up an equation for the volume of the narrow cone, then express $\Delta h_1$ in terms of $h_1:$ \begin{align*} \frac13\pi r_1^2(h_1+\Delta h_1) &= 3\pi h_1+\frac43\pi \\ \frac13 r_1^2(h_1+\Delta h_1) &= 3h_1+\frac43 \\ \frac13\left(\frac{3}{h_1}(h_1+\Delta h_1)\right)^2(h_1+\Delta h_1) &= 3h_1+\frac43 &&\text{by }(1) \\ \frac{3}{h_1^2}(h_1+\Delta h_1)^3 &= 3h_1+\frac43 \\ (h_1+\Delta h_1)^3 &= h_1^3 + \frac{4h_1^2}{9} \\ \Delta h_1 &= \sqrt[3]{h_1^3 + \frac{4h_1^2}{9}}-h_1. \end{align*} Next, we set up an equation for the volume of the wide cone, then express $\Delta h_2$ in terms of $h_2:$ \[\frac13\pi r_2^2(h_2+\Delta h_2) = 12\pi h_2+\frac43\pi.\] Using a similar process from above, we get \[\Delta h_2 = \sqrt[3]{h_2^3+\frac{h_2^2}{9}}-h_2.\] Recall that $\frac{h_1}{h_2}=4.$ Therefore, the requested ratio is \begin{align*} \frac{\Delta h_1}{\Delta h_2}&=\frac{\sqrt[3]{h_1^3 + \frac{4h_1^2}{9}}-h_1}{\sqrt[3]{h_2^3+\frac{h_2^2}{9}}-h_2} \\ &=\frac{\sqrt[3]{(4h_2)^3 + \frac{4(4h_2)^2}{9}}-4h_2}{\sqrt[3]{h_2^3+\frac{h_2^2}{9}}-h_2} \\ &=\frac{\sqrt[3]{4^3\left(h_2^3 + \frac{h_2^2}{9}\right)}-4h_2}{\sqrt[3]{h_2^3+\frac{h_2^2}{9}}-h_2} \\ &=\frac{4\sqrt[3]{h_2^3+\frac{h_2^2}{9}}-4h_2}{\sqrt[3]{h_2^3+\frac{h_2^2}{9}}-h_2} \\ &=\boxed{\textbf{(E) }4:1}. \end{align*} ~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 $3$ and $6$ and infinitely large height. Then the base area of the wide cylinder is $4$ 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 $\boxed{\textbf{(E) } 4}$ times as much.

~scrabbler94

Solution 3 (Calculus)

The volume of the shorter cone can be expressed as $V_1=3\pi h_1$, and the volume of the larger cone can be expressed as $V_2=12\pi h_2$. We also know that the volume changes by $\frac{4\pi}{3}$, because the volume of the $1$cm sphere is $\frac{4\pi}{3}$.

Taking the derivative of the equations, we get: $dV_1=3\pi(dh_1)=\frac{4\pi}{3}$ and $dV_2=12\pi(dh_2)=\frac{4\pi}{3}$.

Therefore, $dh_1=\frac{4\pi}{3}\cdot\frac{1}{3\pi}=\frac{4}{9}$ and $dh_2=\frac{4\pi}{3}\cdot\frac{1}{12\pi}=\frac{1}{9}$. The ratio is $\frac{4}{9}:\frac{1}{9}$ giving us the answer of $\boxed{\textbf{(E) }4:1}$.

~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 $1:4$. We can prove this as follows:

If we fix Cone 2 to have a certain height, then the volume of the cone is $\frac{1}{3}\cdot36\pi{h_1}=12\pi{h_1}$. Now if Cone 1 has the same volume, its height would be $\frac{12\pi{h_1}}{\frac{1}{3}\cdot9\pi}=\frac{12\pi{h_1}}{3\pi}=4h_1$.

Video Solution (Simple and Quick)

https://youtu.be/TgjvviBALac

~ 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)

https://youtu.be/4Iuo7cvGJr8

~ 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

https://youtu.be/c-5-8PnCvCk

~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

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