Difference between revisions of "2006 AMC 10B Problems/Problem 6"
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== Solution == | == Solution == | ||
| − | Since the side of the square is the diameter of the semicircle, the radius of the semicircle is <math> \frac{1}{2}\cdot\frac{2}{\pi}=\frac{1}{\pi} </math> | + | Since the side of the square is the diameter of the semicircle, the radius of the semicircle is <math> \frac{1}{2}\cdot\frac{2}{\pi}=\frac{1}{\pi} </math>. |
| − | Since the length of one of the semicircular arcs is half the circumference of the corresponding circle, the length of one arc is <math> \frac{1}{2}\cdot2\cdot\pi\cdot\frac{1}{\pi}=1</math> | + | Since the length of one of the semicircular arcs is half the circumference of the corresponding circle, the length of one arc is <math> \frac{1}{2}\cdot2\cdot\pi\cdot\frac{1}{\pi}=1</math>. |
Since the desired perimeter is made up of four of these arcs, the perimeter is <math>4\cdot1=4\Rightarrow D</math> | Since the desired perimeter is made up of four of these arcs, the perimeter is <math>4\cdot1=4\Rightarrow D</math> | ||
Revision as of 12:40, 18 July 2006
Problem
A region is bounded by semicircular arcs constructed on the side of a square whose sides measure 
, as shown. What is the perimeter of this region?
Solution
Since the side of the square is the diameter of the semicircle, the radius of the semicircle is 
.
Since the length of one of the semicircular arcs is half the circumference of the corresponding circle, the length of one arc is 
.
Since the desired perimeter is made up of four of these arcs, the perimeter is 
