Difference between revisions of "2014 AMC 8 Problems/Problem 20"
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| − | + | ==Problem== | |
| + | Rectangle <math>ABCD</math> has sides <math>CD=3</math> and <math>DA=5</math>. A circle of radius <math>1</math> is centered at <math>A</math>, a circle of radius <math>2</math> is centered at <math>B</math>, and a circle of radius <math>3</math> is centered at <math>C</math>. Which of the following is closest to the area of the region inside the rectangle but outside all three circles? | ||
| + | |||
| + | <asy> | ||
| + | draw((0,0)--(5,0)--(5,3)--(0,3)--(0,0)); | ||
| + | draw(Circle((0,0),1)); | ||
| + | draw(Circle((0,3),2)); | ||
| + | draw(Circle((5,3),3)); | ||
| + | label("A",(0.2,0),W); | ||
| + | label("B",(0.2,2.8),NW); | ||
| + | label("C",(4.8,2.8),NE); | ||
| + | label("D",(5,0),SE); | ||
| + | label("5",(2.5,0),N); | ||
| + | label("3",(5,1.5),E); | ||
| + | </asy> | ||
| + | |||
| + | <math> \text{(A) }3.5\qquad\text{(B) }4.0\qquad\text{(C) }4.5\qquad\text{(D) }5.0\qquad\text{(E) }5.5 </math> | ||
| + | ==Solution== | ||
| + | The area in the rectangle but outside the circles is the area of the rectangle minus the area of all three of the quarter circles in the rectangle. | ||
| + | |||
| + | The area of the rectangle is <math>3\cdot5 =15</math>. The area of all 3 quarter circles is <math>\frac{\pi}{4}+\frac{\pi(2)^2}{4}+\frac{\pi(3)^2}{4} = \frac{14\pi}{4} = \frac{7\pi}{2}</math>. Therefore the area in the rectangle but outside the circles is <math>15-\frac{7\pi}{2}</math>. <math>\pi</math> is approximately <math>\dfrac{22}{7},</math> and substituting that in will give <math>15-11=\boxed{\text{(B) }4.0}</math> | ||
| + | |||
| + | ==Video Solution== | ||
| + | |||
| + | https://youtu.be/e4gDE2uLQ6A ~DSA_Catachu | ||
| + | |||
| + | https://youtu.be/EPILwH5iA9Q ~savannahsolver | ||
| + | |||
| + | ==Video Solution by OmegaLearn== | ||
| + | https://youtu.be/j3QSD5eDpzU?t=2327 | ||
| + | |||
| + | ~ pi_is_3.14 | ||
| + | |||
| + | ==See Also== | ||
| + | {{AMC8 box|year=2014|num-b=19|num-a=21}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 19:36, 2 January 2023
Problem
Rectangle 
has sides 
and 
. A circle of radius 
is centered at 
, a circle of radius 
is centered at 
, and a circle of radius 
is centered at 
. Which of the following is closest to the area of the region inside the rectangle but outside all three circles?
![[asy] draw((0,0)--(5,0)--(5,3)--(0,3)--(0,0)); draw(Circle((0,0),1)); draw(Circle((0,3),2)); draw(Circle((5,3),3)); label("A",(0.2,0),W); label("B",(0.2,2.8),NW); label("C",(4.8,2.8),NE); label("D",(5,0),SE); label("5",(2.5,0),N); label("3",(5,1.5),E); [/asy]](http://latex.artofproblemsolving.com/e/5/3/e531c5a59c1c4eec0d30c2a3e6ca52cd6bfe139a.png)
Solution
The area in the rectangle but outside the circles is the area of the rectangle minus the area of all three of the quarter circles in the rectangle.
The area of the rectangle is 
. The area of all 3 quarter circles is 
. Therefore the area in the rectangle but outside the circles is 
. 
is approximately 
and substituting that in will give 
Video Solution
https://youtu.be/e4gDE2uLQ6A ~DSA_Catachu
https://youtu.be/EPILwH5iA9Q ~savannahsolver
Video Solution by OmegaLearn
https://youtu.be/j3QSD5eDpzU?t=2327
~ pi_is_3.14
See Also
| 2014 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 19 |
Followed by Problem 21 | |
| 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 AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
