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Find the radius of a circle whose area is equal to the sum of the areas of the two circles of radii $24\ cm$ and $7\ cm$ respectively.
Given: A circle whose area is equal to the sum of the areas of the two circles of radii $24\ cm$ and $7\ cm$ respectively.
To do: To find the radius of the circle.
Solution:
The radius of circles is $r_1=7\ cm,\ r_2=24\ cm$
Area of smaller circle is $A_1=\pi r_{1}^2$
$\Rightarrow A_1=\pi \times 7^2$
$\Rightarrow A_1=49\pi $
Area of Larger circle is $A-2=\pi r_{2}^2$
$\Rightarrow A_2=\pi \times 24^2$
$\Rightarrow A_2=576\pi $
Let $r$ be the radius of the required circle.
Area of required circle $A=\pi r^2$
As given, $A=A_1+A_2$
$\Rightarrow \pi r^2=576\pi +49\pi$
$\Rightarrow r^2=625$
$\Rightarrow r=\sqrt{625}$
$\Rightarrow r=25$
Radius of circle is $25\ cm$.
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