Dumpling wrote:
If the perimeter of square region S and the perimeter of circular region C are equal, then the ratio of the area of S to the area of C is closest to:
A. \(\frac {3}{2}\)
B. \(\frac {4}{3}\)
C. \(\frac {3}{4}\)
D. \(\frac {2}{3}\)
E. \(\frac {1}{2}\)
Let Perimeter of Square \(= 4a\) and Area of Square \(= a^2\)
Let Perimeter of Circle = \(2\)\(\pi\)\(r\) and Area of Circle \(= \pi\)\(r^2\)
Perimeter of Square and Circle are equal.
\(4a =\) \(2\)\(\pi\)\(r\) \(=> a = \frac{r\pi}{2}\)
Ratio of Area of Square to Area of Circle \(= \frac{a^2}{r^2\pi}\)
Substituting value of \("a"\) in above expression we get;
\(\frac{(r\pi/2)^2}{r^2\pi}\) \(= \frac{(r^2\pi^2/4)}{r^2\pi}\) \(= \frac{r^2\pi^2}{4r^2\pi}\)
Hence Ratio of Area of Square to Area of Circle = \(\frac{\pi}{4}\)
\(\pi\) is approx \(= 3\)
Hence required ratio \(= \frac{3}{4}\)
Answer C