Difference between revisions of "2014 AMC 8 Problems/Problem 20"
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==Problem== | ==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? | 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> | <asy> | ||
draw((0,0)--(5,0)--(5,3)--(0,3)--(0,0)); | draw((0,0)--(5,0)--(5,3)--(0,3)--(0,0)); | ||
Line 13: | Line 14: | ||
label("3",(5,1.5),E); | label("3",(5,1.5),E); | ||
</asy> | </asy> | ||
− | <math> \ | + | |
+ | <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== | ==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 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. | ||
Line 22: | Line 24: | ||
https://youtu.be/e4gDE2uLQ6A ~DSA_Catachu | 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== | ==See Also== | ||
{{AMC8 box|year=2014|num-b=19|num-a=21}} | {{AMC8 box|year=2014|num-b=19|num-a=21}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 20:36, 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?
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.