Question

Find the breadth of the ring if the area of two concentric circles forming a ring are \[154c{m^2}\] and \[616c{m^2}\]? Choose the correct answer
(A) $21cm$
(B) $14cm$
(C) $56cm$
(D) $7cm$

Answer 1

Hint: Here we have a ring, outer and inner area is given by those areas we will find outer and inner radius by applying area formula, after getting both radii we will subtract those and find breadth of ring.

Complete step-by-step answer:

According to the question it is given that the area of the outer circle is
$ \Rightarrow Area{\text{ }}of{\text{ }}the{\text{ }}outer{\text{ }}circle{\text{ = 616c}}{{\text{m}}^2}$
And the area of the inner circle is
$ \Rightarrow Area{\text{ }}of{\text{ }}the{\text{ inne}}r{\text{ }}circle{\text{ = 154c}}{{\text{m}}^2}$
We will find outer radius from here by area formula
$\therefore Outer{\text{ }}radius = \sqrt {\dfrac{{Area}}{\pi }} $
Now we put the value of area and $\pi = \dfrac{{22}}{7}$ in the formula, we get
$Outer{\text{ }}radius = \sqrt {\dfrac{{616}}{{\dfrac{{22}}{7}}}} $
Now $7$ is in the denominator it goes into the numerator and we multiply it by $616$, we get
$ \Rightarrow \sqrt {\dfrac{{616 \times 7}}{{22}}} cm$
By solving the above equation we get
$ = \sqrt {196} $
Now by solving the square root we get the outer radius as
$ = 14cm$
Now the area of the inner circle is $154c{m^2}$, We will find inner radius by using area formula
$\therefore Inner{\text{ }}radius = \sqrt {\dfrac{{Area}}{\pi }} $
Now we put the value of area and $\pi = \dfrac{{22}}{7}$ in the formula, we get
$Inner{\text{ }}radius = \sqrt {\dfrac{{154}}{{\dfrac{{22}}{7}}}} $
Now $7$ is in the denominator it goes into the numerator and we multiply it by $154$, we get
$ \Rightarrow \sqrt {\dfrac{{154 \times 7}}{{22}}} cm$
By solving the above equation we get
$ \Rightarrow \sqrt {49} $
Now by solving the square root we get the inner radius as
$ = 7cm$
Here we get outer and inner radius by subtracting those radius we will find breadth of ring
$\therefore Breadth{\text{ }}of{\text{ }}the{\text{ }}ring{\text{ }}so{\text{ }}formed = {\text{ }}outer{\text{ }}radius{\text{ }}-{\text{ }}inner{\text{ }}radius$
For finding the breadth of the ring we put the values of outer and inner radius in the formula we get,
 $ \Rightarrow Breadth{\text{ }}of{\text{ }}the{\text{ }}ring{\text{ }}so{\text{ }}formed = 14 - 7$
By solving equation we get,
$ \Rightarrow Breadth{\text{ }}of{\text{ }}the{\text{ }}ring{\text{ }}so{\text{ }}formed = 7$
So, the breadth of the ring is $7cm$.

Hence the option D is correct.

Note: Here we can apply short formula that is $Breadth{\text{ = }}\sqrt {\dfrac{{{\text{area of outer circle}}}}{\pi } - \dfrac{{{\text{area of inner circle}}}}{\pi }} $ by using this formula we simply find breadth of ring.