Difference between revisions of "2004 AMC 12B Problems/Problem 7"
(→Solution) |
|||
| (4 intermediate revisions by 3 users not shown) | |||
| Line 2: | Line 2: | ||
== Problem == | == Problem == | ||
| − | A square has sides of length 10, and a circle centered at one of its vertices has radius 10. What is the area of the union of the regions enclosed by the square and the circle? | + | A square has sides of length <math>10</math>, and a circle centered at one of its vertices has radius <math>10</math>. What is the area of the union of the regions enclosed by the square and the circle? |
| + | |||
| + | <math>\mathrm{(A)\ }200+25\pi\quad\mathrm{(B)\ }100+75\pi\quad\mathrm{(C)\ }75+100\pi\quad\mathrm{(D)\ }100+100\pi\quad\mathrm{(E)\ }100+125\pi</math> | ||
| + | |||
| + | == Video Solution 1== | ||
| + | https://youtu.be/IGN4XxJIbE0 | ||
| + | |||
| + | ~Education, the Study of Everything | ||
| + | |||
| − | |||
== Solution == | == Solution == | ||
| − | The area of the circle is <math>S_{\bigcirc}=100\pi</math> | + | The area of the circle is <math>S_{\bigcirc}=100\pi</math>; the area of the square is <math>S_{\square}=100</math>. |
| − | Exactly <math>1 | + | Exactly <math>\frac{1}{4}</math> of the circle lies inside the square. Thus the total area is <math>\dfrac34 S_{\bigcirc}+S_{\square}=\boxed{\mathrm{(B)\ }100+75\pi}</math>. |
<asy> | <asy> | ||
| Line 24: | Line 31: | ||
{{AMC12 box|year=2004|ab=B|num-b=6|num-a=8}} | {{AMC12 box|year=2004|ab=B|num-b=6|num-a=8}} | ||
{{AMC10 box|year=2004|ab=B|num-b=8|num-a=10}} | {{AMC10 box|year=2004|ab=B|num-b=8|num-a=10}} | ||
| + | |||
| + | [[Category:Introductory Geometry Problems]] | ||
| + | [[Category:Area Problems]] | ||
| + | {{MAA Notice}} | ||
Latest revision as of 18:23, 22 October 2022
- The following problem is from both the 2004 AMC 12B #7 and 2004 AMC 10B #9, so both problems redirect to this page.
Contents
Problem
A square has sides of length 
, and a circle centered at one of its vertices has radius . What is the area of the union of the regions enclosed by the square and the circle?
Video Solution 1
~Education, the Study of Everything
Solution
The area of the circle is 
; the area of the square is 
.
Exactly 
of the circle lies inside the square. Thus the total area is 
.
![[asy] Draw(Circle((0,0),10)); Draw((0,0)--(10,0)--(10,10)--(0,10)--(0,0)); label("$10$",(5,0),S); label("$10$",(0,5),W); dot((0,0)); [/asy]](http://latex.artofproblemsolving.com/9/f/c/9fcea2965eceab15b23e988726a4b3446b01b59f.png)
See Also
| 2004 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 6 |
Followed by Problem 8 |
| 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 | |
| 2004 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 8 |
Followed by Problem 10 | |
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
