monirjewel wrote:

In the figure shown, if the area of the shaded region is 3 times the area of the smaller circular region, then the circumference of the larger circle is how many times the circumference of the smaller circle?
A. \(4\)
B. \(3\)
C. \(2\)
D. \(\sqrt{3}\)
E. \(\sqrt{2}\)
Attachment:
1111220830.jpg
AlgebraicallyLet a = Small circle's area
Let s = Shaded region's area
Let A = Large circle's area
\(s = 3a\)
\(A = s + a\)
\(A = 3a + a\)
\(A = 4a\)
\(a = \pi r^2\)
\(A = \pi R^2\)From above:
\(A = 4a\)
\(\pi R^2 = 4\pi r^2\)
\(R^2 = 4r^2\)
\(\sqrt{R^2}=\sqrt{4r^2}\)
\(R = 2r\)*
Circumference, Large and Small
C =
\(2\pi R\), and c =
\(2\pi r\)\(R = 2r\), so
\(\frac{C}{c}=\frac{2\pi (2r)}{2\pi r}= \frac{2r}{r}=2\)The large circle's circumference is two times the small circle's circumference.
Answer C
Numbers and algebraLarge circle's area: Small circle's area?Small circle's area = x
Shaded region's area = 3x
Large circle's area = 3x + x =
4xLarge circle circumference/ small circle's circumference?Let Small circle's radius
\(r = 1\)Area of Small:
\(\pi r^2 = \pi\)Area of Large = (
4x) =
\((4*\pi) = 4\pi\)Radius, R, of Large circle:
\(4\pi =\pi R^2\)
\(R = 2\)Circumference, Small:
\(2 \pi r = 2 \pi\)Circumference, Large:
\(2 \pi R = 4 \pi\)\(\frac{Large}{Small}=\frac{4\pi}{2\pi} = 2\) The large circle's circumference is two times the small circle's circumference.
Answer C
**
OR
\(A = 3a + a\)
\(A - 3a = a\) AND \(A = \pi R^2\)
\(\pi R^2 - 3\pi r^2=\pi r^2\)
\(\pi (R^2 - 3r^2)=\pi r^2\)
\(R^2 - 3r^2 = r^2\)
\(R^2 = 4r^2\) Take square roots:
\(R = 2r\) _________________