Find the area enclosed between two concentric circles of radii 3.5 cm and 7 cm. A third concentric circle is drawn outside the 7 cm circle, such that the area enclosed between it and the 7 cm circle is same as that between the two inner circles. Find the radius of the third circle correct to one decimal place.

Given:

Radius of first circle $r_1 = 3.5\ cm$

Radius of second circle $r_2 = 7\ cm$

A third concentric circle is drawn outside the 7 cm circle, such that the area enclosed between it and the 7 cm circle is same as that between the two inner circles.

To do:

We have to find the area enclosed between two concentric circles of radii 3.5 cm and 7 cm and the radius of the third circle correct to one decimal place.

Solution:

Let the radius of the third circle be $r_3$.

Area between the two concentric circles $=\pi(r_{2}^{2}-r_{1}^{2})$

$=\pi(7^{2}-(3.5)^{2}) \mathrm{cm}^{2}$

$=\frac{22}{7}(49-12.25) \mathrm{cm}^{2}$

$=\frac{22}{7} \times 36.75 \mathrm{~cm}^{2}$

$=115.5 \mathrm{~cm}^{2}$

Area between the second and the third circle $=\pi (r_3^{2}-7^{2})$

$=\frac{22}{7}(r_3^2-49)$

$=\frac{22}{7} r_3^2-154$

The area between first and second circle is same as that between the two inner circles.

Therefore,

$\frac{22}{7} r_3^{2}-154=115.5$

$\Rightarrow \frac{22}{7} r_3^{2}=115.5+154.0$

$\Rightarrow \frac{22}{7} r_3^{2}=269.5$

$\Rightarrow r_3^{2}=\frac{296.5 \times 7}{22}$

$\Rightarrow r_3^{2}=\frac{24.5 \times 7}{2}$

$=\frac{171.5}{2}$

$=85.75$

$\Rightarrow r_3=\sqrt{85.75}$

$=9.26$

The radius of the third circle is $9.26 \mathrm{~cm}$.

Tutorialspoint

Simply Easy Learning

Updated on: 10-Oct-2022

71 Views

Kickstart Your Career

Get certified by completing the course

Get Started