question_answer

A circular path is made from two concentric circular rings in such a way that the smaller ring when allowed to roll over the circumference of the bigger ring. it takes three full revolutions. If the area of the pathway is equal to n times the area of the smaller ring. then n is equal to

A)  4

B)  6

C)  8

D)  10

Correct Answer: C

Solution :

  Radius of smaller ring = r units Radius of bigger ring = R units According to the question, \[2\pi R=3\times 2\pi r\] \[\Rightarrow \] \[R=3r\] Area of shaded part (path) \[=\pi {{R}^{2}}-\pi {{r}^{2}}=\pi {{(3r)}^{2}}-\pi {{r}^{2}}\] \[=9\pi {{r}^{2}}-\pi {{r}^{2}}=8\pi {{r}^{2}}\] \[\therefore \] \[8\pi {{r}^{2}}=n\,.\,n{{r}^{2}}\] \[\Rightarrow \] \[n=\frac{8\pi {{r}^{2}}}{\pi {{r}^{2}}}=8\]