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Difference between revisions of "2015 AMC 10A Problems/Problem 14"

(+diagram, format)
(added long solution)
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==Solution 3==
 
==Solution 3==
 
The arrow travels a path of radius 30 (20 from the interior clock and 10 from the radius of the disk itself). We note that 1 complete rotation of 360 degrees is needed for the arrow to appear up again, so therefore, the disk must travel its circumference before the arrow goes up. Its circumference is <math>20\pi</math>, so that is <math>20\pi</math> traveled on a <math>60\pi</math> arrow path. This is a ratio of 1/3, so the angle it carves is 120 degrees, which leads us to the correct answer of  4 o' clock. <math>\boxed{\textbf{(C)}}</math>
 
The arrow travels a path of radius 30 (20 from the interior clock and 10 from the radius of the disk itself). We note that 1 complete rotation of 360 degrees is needed for the arrow to appear up again, so therefore, the disk must travel its circumference before the arrow goes up. Its circumference is <math>20\pi</math>, so that is <math>20\pi</math> traveled on a <math>60\pi</math> arrow path. This is a ratio of 1/3, so the angle it carves is 120 degrees, which leads us to the correct answer of  4 o' clock. <math>\boxed{\textbf{(C)}}</math>
 +
 +
==Solution 4==
 +
Suppose that the small disk also had a clock face on it, and that both disks were toothed wheels, free to rotate around their centers. The part of the picture where they engage would look like this:
 +
<asy>
 +
fill(arc((0,0),2,30,150)--cycle,lightgrey);
 +
draw(arc((0,0),2,30,150));
 +
draw(1.8*dir(90)--2*dir(90));
 +
draw(1.8*dir(60)--2*dir(60));
 +
label("12",1.56*dir(90));
 +
label("1",1.56*dir(60));
 +
 +
draw(arc((0,3),1,-15,-165));
 +
draw(0.9*dir(-90)+(0,3)--dir(-90)+(0,3));
 +
draw(0.9*dir(-60)+(0,3)--dir(-60)+(0,3));
 +
draw(0.9*dir(-30)+(0,3)--dir(-30)+(0,3));
 +
label("6",.74*dir(-90)+(0,3));
 +
label("5",.74*dir(-60)+(0,3));
 +
label("4",.74*dir(-30)+(0,3));
 +
</asy>
 +
The small cog has half the radius, and therefore half the circumference. If the large cog turns <math>30^\circ</math> anticlockwise (i.e. 1 hour), the small cog turns <math>60^\circ</math> clockwise (i.e. 2 hours).
 +
<asy>
 +
fill(arc((0,0),2,30,150)--cycle,lightgrey);
 +
draw(arc((0,0),2,30,150),EndArcArrow);label("$30^\circ$",2*dir(150),W);
 +
draw(1.8*dir(120)--2*dir(120));
 +
draw(1.8*dir(90)--2*dir(90));
 +
label(rotate(30)*"12",1.56*dir(120));
 +
label(rotate(30)*"1",1.56*dir(90));
 +
 +
draw(arc((0,3),1,-15,-165),EndArcArrow);label("$60^\circ$",dir(-165)+(0,3),W);
 +
draw(0.9*dir(-90)+(0,3)--dir(-90)+(0,3));
 +
draw(0.9*dir(-120)+(0,3)--dir(-120)+(0,3));
 +
draw(0.9*dir(-150)+(0,3)--dir(-150)+(0,3));
 +
label(rotate(-60)*"6",.74*dir(-150)+(0,3));
 +
label(rotate(-60)*"5",.74*dir(-120)+(0,3));
 +
label(rotate(-60)*"4",.74*dir(-90)+(0,3));
 +
</asy>
 +
However, in the original problem the large cog does not rotate; it stays where it is. Therefore we must turn the whole diagram above <math>30^\circ</math> clockwise to see what happens when the small cog rolls around it.
 +
<asy>
 +
fill(arc((0,0),2,30,150)--cycle,lightgrey);
 +
draw(arc((0,0),2,30,150));
 +
draw(1.8*dir(90)--2*dir(90));
 +
draw(1.8*dir(60)--2*dir(60));
 +
label("12",1.56*dir(90));
 +
label("1",1.56*dir(60));
 +
 +
pair c=(1.5,sqrt(27)/2);
 +
draw(arc(c,1,0,-200),EndArcArrow);label("$90^\circ$",dir(-180)+c,W);
 +
draw(0.9*dir(-120)+c--dir(-120)+c);
 +
draw(0.9*dir(-150)+c--dir(-150)+c);
 +
draw(0.9*dir(-180)+c--dir(-180)+c);
 +
label(rotate(-90)*"6",.74*dir(-180)+c);
 +
label(rotate(-90)*"5",.74*dir(-150)+c);
 +
label(rotate(-90)*"4",.74*dir(-120)+c);
 +
</asy>
 +
It turns out that, when the point of tangency moves <math>30^\circ</math> clockwise (one hour), from our point of view the small disk rotates <math>90^\circ</math> clockwise (three hours). Thus, for the small disk to perform a complete rotation (twelve hours) from our point of view, the point of tangency must move round four hours. So the answer is <math>\boxed{\textbf{(C) }\text{4 o' clock}}</math>
  
 
==See Also==
 
==See Also==

Revision as of 05:29, 15 February 2016


The diagram below shows the circular face of a clock with radius $20$ cm and a circular disk with radius $10$ cm externally tangent to the clock face at $12$ o' clock. The disk has an arrow painted on it, initially pointing in the upward vertical direction. Let the disk roll clockwise around the clock face. At what point on the clock face will the disk be tangent when the arrow is next pointing in the upward vertical direction? [asy]size(170);defaultpen(linewidth(0.9)+fontsize(13pt));draw(unitcircle^^circle((0,1.5),0.5)); path arrow = origin--(-0.13,-0.35)--(-0.06,-0.35)--(-0.06,-0.7)--(0.06,-0.7)--(0.06,-0.35)--(0.13,-0.35)--cycle; for(int i=1;i<=12;i=i+1){draw(0.9*dir(90-30*i)--dir(90-30*i));label("$"+(string) i+"$",0.78*dir(90-30*i));} dot(origin);draw(shift((0,1.87))*arrow);draw(arc(origin,1.5,68,30),EndArrow(size=12));[/asy]

$\textbf{(A) }\text{2 o' clock} \qquad\textbf{(B) }\text{3 o' clock} \qquad\textbf{(C) }\text{4 o' clock} \qquad\textbf{(D) }\text{6 o' clock} \qquad\textbf{(E) }\text{8 o' clock}$

Solution 1

The circumference of the clock is twice that of the disk. So, a quarter way around the clock (3:00), the point halfway around the disk will be tangent. The arrow will point to the left. We can see the disk made a 75% rotation from 12 to 3, and 3 is 75% of 4, so it would make 100% rotation from 12 to 4. The answer is $\boxed{\textbf{(C) }4 \ \text{o' clock}}$ .

Solution 2

The rotation factor of the arrow is the sum of the rates of the regular rotation of the arrow (360° every 360° rotation = 1) and the rotation of the disk around the clock with twice the circumference (360° every 180° = 2). Thus, the rotation factor of the arrow is 3, and so our answer corresponds to 360°/3 = 120°, which is 4 o' clock. $\boxed{\textbf{(C)}}$

Solution 3

The arrow travels a path of radius 30 (20 from the interior clock and 10 from the radius of the disk itself). We note that 1 complete rotation of 360 degrees is needed for the arrow to appear up again, so therefore, the disk must travel its circumference before the arrow goes up. Its circumference is $20\pi$, so that is $20\pi$ traveled on a $60\pi$ arrow path. This is a ratio of 1/3, so the angle it carves is 120 degrees, which leads us to the correct answer of 4 o' clock. $\boxed{\textbf{(C)}}$

Solution 4

Suppose that the small disk also had a clock face on it, and that both disks were toothed wheels, free to rotate around their centers. The part of the picture where they engage would look like this: [asy] fill(arc((0,0),2,30,150)--cycle,lightgrey); draw(arc((0,0),2,30,150)); draw(1.8*dir(90)--2*dir(90)); draw(1.8*dir(60)--2*dir(60)); label("12",1.56*dir(90)); label("1",1.56*dir(60)); draw(arc((0,3),1,-15,-165)); draw(0.9*dir(-90)+(0,3)--dir(-90)+(0,3)); draw(0.9*dir(-60)+(0,3)--dir(-60)+(0,3)); draw(0.9*dir(-30)+(0,3)--dir(-30)+(0,3)); label("6",.74*dir(-90)+(0,3)); label("5",.74*dir(-60)+(0,3)); label("4",.74*dir(-30)+(0,3)); [/asy] The small cog has half the radius, and therefore half the circumference. If the large cog turns $30^\circ$ anticlockwise (i.e. 1 hour), the small cog turns $60^\circ$ clockwise (i.e. 2 hours). [asy] fill(arc((0,0),2,30,150)--cycle,lightgrey); draw(arc((0,0),2,30,150),EndArcArrow);label("$30^\circ$",2*dir(150),W); draw(1.8*dir(120)--2*dir(120)); draw(1.8*dir(90)--2*dir(90)); label(rotate(30)*"12",1.56*dir(120)); label(rotate(30)*"1",1.56*dir(90)); draw(arc((0,3),1,-15,-165),EndArcArrow);label("$60^\circ$",dir(-165)+(0,3),W); draw(0.9*dir(-90)+(0,3)--dir(-90)+(0,3)); draw(0.9*dir(-120)+(0,3)--dir(-120)+(0,3)); draw(0.9*dir(-150)+(0,3)--dir(-150)+(0,3)); label(rotate(-60)*"6",.74*dir(-150)+(0,3)); label(rotate(-60)*"5",.74*dir(-120)+(0,3)); label(rotate(-60)*"4",.74*dir(-90)+(0,3)); [/asy] However, in the original problem the large cog does not rotate; it stays where it is. Therefore we must turn the whole diagram above $30^\circ$ clockwise to see what happens when the small cog rolls around it. [asy] fill(arc((0,0),2,30,150)--cycle,lightgrey); draw(arc((0,0),2,30,150)); draw(1.8*dir(90)--2*dir(90)); draw(1.8*dir(60)--2*dir(60)); label("12",1.56*dir(90)); label("1",1.56*dir(60)); pair c=(1.5,sqrt(27)/2); draw(arc(c,1,0,-200),EndArcArrow);label("$90^\circ$",dir(-180)+c,W); draw(0.9*dir(-120)+c--dir(-120)+c); draw(0.9*dir(-150)+c--dir(-150)+c); draw(0.9*dir(-180)+c--dir(-180)+c); label(rotate(-90)*"6",.74*dir(-180)+c); label(rotate(-90)*"5",.74*dir(-150)+c); label(rotate(-90)*"4",.74*dir(-120)+c); [/asy] It turns out that, when the point of tangency moves $30^\circ$ clockwise (one hour), from our point of view the small disk rotates $90^\circ$ clockwise (three hours). Thus, for the small disk to perform a complete rotation (twelve hours) from our point of view, the point of tangency must move round four hours. So the answer is $\boxed{\textbf{(C) }\text{4 o' clock}}$

See Also

2015 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 13 		Followed by
Problem 15
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

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