Question

A circular wire of radius 1 dm is cut and is placed along the circumference of a circle of radius of one meter. The angle subtended by the wire at the center of the circle is equal to
A. $\dfrac{\pi }{4}$ radian
B. $\dfrac{\pi }{3}$ radian
C. $\dfrac{\pi }{5}$ radian
D. $\dfrac{\pi }{{10}}$ radian

Answer 1

Hint:
Firstly we have to calculate the circumference of both the circles, smaller and larger both. Then we have an arc and angle subtended at the center formula with the help of which we can easily calculate the required quantity, here we have to calculate angle subtended by the small wire at center.

Complete step by step solution:
Given:
 r = 1 dm
 R = 1 m
Circumference of smaller wire circle = $2\pi r$
      also we can call it = $2 \times \pi \times 0.1$
  Length of smaller wire = $0.2\pi $m = Arc1
Circumference of larger wire circle = $2\pi R$
    Also we can call it = $2\pi \times 1$
   Length of larger arc = $2\pi $ = Arc2
Let angle subtended by the smaller wire be $\theta $, which is required.
Since the complete angle which can be subtended on a circle’s center is ${360^ \circ }$, which is subtended by the largest arc possible that is the circumference of the circle.
$\therefore $ The ratio of the angle $\theta $ subtended is directly proportional to the ratio of Arcs
$\theta $ = $\dfrac{{Arc1}}{{Arc2}} \times {360^ \circ }$
$\theta $ = $\dfrac{{0.2\pi }}{{2\pi }} \times {360^ \circ }$
$\theta $ = $\dfrac{{{{360}^ \circ }}}{{{{10}^ \circ }}}$
$\theta $ = ${36^ \circ }$
$\theta $ = ${36^ \circ }$$ \times \dfrac{\pi }{{{{180}^ \circ }}}$
$\theta $ = $\dfrac{\pi }{5}$ radian
Hence, option C is correct

Note:
Change all the dimensions in the same unit before doing any calculations.
1 m = 10 dm
${180^ \circ } = \pi $