Difference between revisions of "2019 AMC 10A Problems/Problem 16"
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<asy>unitsize(20);filldraw(circle((0,0),2*sqrt(3)+1),rgb(0.5,0.5,0.5));filldraw(circle((-2,0),1),white);filldraw(circle((0,0),1),white);filldraw(circle((2,0),1),white);filldraw(circle((1,sqrt(3)),1),white);filldraw(circle((3,sqrt(3)),1),white);filldraw(circle((-1,sqrt(3)),1),white);filldraw(circle((-3,sqrt(3)),1),white);filldraw(circle((1,-1*sqrt(3)),1),white);filldraw(circle((3,-1*sqrt(3)),1),white);filldraw(circle((-1,-1*sqrt(3)),1),white);filldraw(circle((-3,-1*sqrt(3)),1),white);filldraw(circle((0,2*sqrt(3)),1),white);filldraw(circle((0,-2*sqrt(3)),1),white);</asy> | <asy>unitsize(20);filldraw(circle((0,0),2*sqrt(3)+1),rgb(0.5,0.5,0.5));filldraw(circle((-2,0),1),white);filldraw(circle((0,0),1),white);filldraw(circle((2,0),1),white);filldraw(circle((1,sqrt(3)),1),white);filldraw(circle((3,sqrt(3)),1),white);filldraw(circle((-1,sqrt(3)),1),white);filldraw(circle((-3,sqrt(3)),1),white);filldraw(circle((1,-1*sqrt(3)),1),white);filldraw(circle((3,-1*sqrt(3)),1),white);filldraw(circle((-1,-1*sqrt(3)),1),white);filldraw(circle((-3,-1*sqrt(3)),1),white);filldraw(circle((0,2*sqrt(3)),1),white);filldraw(circle((0,-2*sqrt(3)),1),white);</asy> | ||
− | <math>\textbf{(A) } 4 \pi \sqrt{3} \qquad\textbf{(B) } 7 \pi \qquad\textbf{(C) } \pi(3\sqrt{3} +2) \qquad\textbf{(D) } 10 \pi (\sqrt{3} - 1) \qquad\textbf{(E) } \pi(\sqrt{3} + 6)</math> | + | <math>\textbf{(A) } 4 \pi \sqrt{3} \qquad\textbf{(B) } 7 \pi \qquad\textbf{(C) } \pi\left(3\sqrt{3} +2\right) \qquad\textbf{(D) } 10 \pi \left(\sqrt{3} - 1\right) \qquad\textbf{(E) } \pi\left(\sqrt{3} + 6\right)</math> |
==Solution 1== | ==Solution 1== | ||
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unitsize(20);filldraw(circle((0,0),2*sqrt(3)+1),rgb(0.5,0.5,0.5));filldraw(circle((-2,0),1),white);filldraw(circle((0,0),1),white);filldraw(circle((2,0),1),white);filldraw(circle((1,sqrt(3)),1),white);filldraw(circle((3,sqrt(3)),1),white);filldraw(circle((-1,sqrt(3)),1),white);filldraw(circle((-3,sqrt(3)),1),white);filldraw(circle((1,-1*sqrt(3)),1),white);filldraw(circle((3,-1*sqrt(3)),1),white);filldraw(circle((-1,-1*sqrt(3)),1),white);filldraw(circle((-3,-1*sqrt(3)),1),white);filldraw(circle((0,2*sqrt(3)),1),white);filldraw(circle((0,-2*sqrt(3)),1),white); | unitsize(20);filldraw(circle((0,0),2*sqrt(3)+1),rgb(0.5,0.5,0.5));filldraw(circle((-2,0),1),white);filldraw(circle((0,0),1),white);filldraw(circle((2,0),1),white);filldraw(circle((1,sqrt(3)),1),white);filldraw(circle((3,sqrt(3)),1),white);filldraw(circle((-1,sqrt(3)),1),white);filldraw(circle((-3,sqrt(3)),1),white);filldraw(circle((1,-1*sqrt(3)),1),white);filldraw(circle((3,-1*sqrt(3)),1),white);filldraw(circle((-1,-1*sqrt(3)),1),white);filldraw(circle((-3,-1*sqrt(3)),1),white);filldraw(circle((0,2*sqrt(3)),1),white);filldraw(circle((0,-2*sqrt(3)),1),white); | ||
− | pair O,A,B,C; | + | pair O,A,B,C,H; |
O=(0,0); | O=(0,0); | ||
A=(-1,sqrt(3)); | A=(-1,sqrt(3)); | ||
B=(1,sqrt(3)); | B=(1,sqrt(3)); | ||
C=(0,sqrt(3)*2); | C=(0,sqrt(3)*2); | ||
+ | H=(0,sqrt(3)); | ||
draw(O--A); | draw(O--A); | ||
draw(A--C); | draw(A--C); | ||
Line 32: | Line 33: | ||
label("B",B,E); | label("B",B,E); | ||
label("C",C, N); | label("C",C, N); | ||
+ | label("H",H, NE); | ||
</asy> | </asy> | ||
− | In the diagram above, notice that triangle <math>OAB</math> and triangle <math>ABC</math> are congruent and equilateral with side length <math>2</math>. We can see the radius of the larger circle is | + | In the diagram above, notice that triangle <math>OAB</math> and triangle <math>ABC</math> are congruent and equilateral with side length <math>2</math>. We can see the radius of the larger circle is <math>2\overline{OH} + 1</math>. Using <math>30^{\circ}-60^{\circ}-90^{\circ}</math> triangles, we know <math>\overline{OH} = \sqrt{3}</math>. Therefore, the radius of the larger circle is <math>2\sqrt{3}+1</math>. |
− | The area of the larger circle is thus <math>(2\sqrt{3}+1)^2 \pi = (13+4\sqrt{3})\pi</math>, and the sum of the areas of the smaller circles is <math>13\pi</math>, so the area of the dark region is <math>(13+4\sqrt{3})\pi-13\pi = | + | The area of the larger circle is thus <math>\left(2\sqrt{3}+1\right)^2 \pi = \left(13+4\sqrt{3}\right)\pi</math>, and the sum of the areas of the smaller circles is <math>13\pi</math>, so the area of the dark region is <math>\left(13+4\sqrt{3}\right)\pi-13\pi = \boxed{\textbf{(A) } 4 \pi \sqrt{3}}</math>. |
==Solution 2== | ==Solution 2== | ||
− | We can form an equilateral triangle with side length <math>6</math> from the centers of three of the unit circles tangent to the outer circle. The radius of the outer circle is the circumradius of the triangle plus <math>1</math>. By using <math>R = \frac{abc}{ | + | We can form an equilateral triangle with side length <math>6</math> from the centers of three of the unit circles tangent to the outer circle. The radius of the outer circle is the circumradius of the triangle plus <math>1</math>. By using <math>R = \frac{abc}{4K}</math> or <math>R=\frac{a}{2\sin{A}}</math>, we get the radius as <math>\frac{6}{\sqrt{3}}+1</math>. |
− | The shaded | + | The shaded area is thus <math>\pi((\frac{6}{\sqrt{3}}+1)^2-13) = \boxed{\textbf{(A) } 4 \pi \sqrt{3}}</math>. |
==Solution 3== | ==Solution 3== | ||
− | Like in Solution 2, we can form an equilateral triangle with side length <math>6</math> from the centers of three of the unit circles tangent to the outer circle. We can find the height of this triangle to be <math>3\sqrt{3}</math>. Then, we can form another equilateral triangle from the centers of the second and third circles in the third row and the center of the bottom circle with side length <math>2</math>. The height of this triangle is clearly <math>\sqrt{3}</math>. Therefore the diameter of the large circle is <math>4\sqrt{3} + 2</math> and the radius is <math>\frac{4\sqrt{3}+2}{2} = 2\sqrt{3} + 1</math>. The area is thus <math> | + | Like in Solution 2, we can form an equilateral triangle with side length <math>6</math> from the centers of three of the unit circles tangent to the outer circle. We can find the height of this triangle to be <math>3\sqrt{3}</math>. Then, we can form another equilateral triangle from the centers of the second and third circles in the third row and the center of the bottom circle with side length <math>2</math>. The height of this triangle is clearly <math>\sqrt{3}</math>. Therefore the diameter of the large circle is <math>4\sqrt{3} + 2</math> and the radius is <math>\frac{4\sqrt{3}+2}{2} = 2\sqrt{3} + 1</math>. The area of the large circle is thus <math>\pi\left(2\sqrt{3} + 1\right)^{2} = \pi \cdot \left(13 + 4\sqrt{3}\right) = 13\pi + 4\pi\sqrt{3}</math>. The total area of the <math>13</math> smaller circles is <math>13\pi</math>, so the shaded area is <math>\left(13\pi + 4\pi\sqrt{3}\right) - 13\pi = \boxed{\textbf{(A) } 4 \pi \sqrt{3}}</math>. |
==Solution 4== | ==Solution 4== | ||
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</asy> | </asy> | ||
− | In the diagram above, <math>AB=4</math> | + | In the diagram above, <math>AB=4</math> and <math>BC=2</math>, so <math>AC=\sqrt{4^2-2^2}=2\sqrt{3}</math>. The larger circle's radius is <math>AC+1=2\sqrt{3}+1</math>, so the larger circle's area is <math>\pi\left(2\sqrt{3}+1\right)^2=\pi\left(13+4\sqrt{3}\right)=13\pi+4\pi\sqrt{3}</math>. Now, subtracting the combined area of the smaller circles gives <math>13\pi+4\pi\sqrt{3}-13\pi=\boxed{\textbf{(A) } 4 \pi \sqrt{3}}</math>. |
+ | |||
+ | ==Video Solution1== | ||
+ | https://youtu.be/BBoWwpToBZ8 | ||
+ | |||
+ | Education, the Study of Everything | ||
+ | |||
+ | ==Video Solution 2 by SpreadTheMathLove== | ||
+ | https://www.youtube.com/watch?v=q9L66I9f9nI | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/WOz6CpF-6mI | ||
+ | |||
+ | ~savannahsolver | ||
+ | |||
+ | == Video Solution by OmegaLearn == | ||
+ | https://youtu.be/NsQbhYfGh1Q?t=1793 | ||
+ | |||
+ | ~ pi_is_3.14 | ||
==See Also== | ==See Also== |
Latest revision as of 08:20, 20 November 2023
- The following problem is from both the 2019 AMC 10A #16 and 2019 AMC 12A #10, so both problems redirect to this page.
Contents
Problem
The figure below shows circles of radius
within a larger circle. All the intersections occur at points of tangency. What is the area of the region, shaded in the figure, inside the larger circle but outside all the circles of radius
Solution 1
In the diagram above, notice that triangle and triangle
are congruent and equilateral with side length
. We can see the radius of the larger circle is
. Using
triangles, we know
. Therefore, the radius of the larger circle is
.
The area of the larger circle is thus , and the sum of the areas of the smaller circles is
, so the area of the dark region is
.
Solution 2
We can form an equilateral triangle with side length from the centers of three of the unit circles tangent to the outer circle. The radius of the outer circle is the circumradius of the triangle plus
. By using
or
, we get the radius as
.
The shaded area is thus .
Solution 3
Like in Solution 2, we can form an equilateral triangle with side length from the centers of three of the unit circles tangent to the outer circle. We can find the height of this triangle to be
. Then, we can form another equilateral triangle from the centers of the second and third circles in the third row and the center of the bottom circle with side length
. The height of this triangle is clearly
. Therefore the diameter of the large circle is
and the radius is
. The area of the large circle is thus
. The total area of the
smaller circles is
, so the shaded area is
.
Solution 4
In the diagram above, and
, so
. The larger circle's radius is
, so the larger circle's area is
. Now, subtracting the combined area of the smaller circles gives
.
Video Solution1
Education, the Study of Everything
Video Solution 2 by SpreadTheMathLove
https://www.youtube.com/watch?v=q9L66I9f9nI
Video Solution
~savannahsolver
Video Solution by OmegaLearn
https://youtu.be/NsQbhYfGh1Q?t=1793
~ pi_is_3.14
See Also
2019 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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 |
2019 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.