VerifiedHint: First of all, find the area of the first circle and second circle. Then consider the radius of the required circle as a variable and find its area. Then equate the required circle area to the sum of the areas of first and second circle which will lead to the solution of the given problem.
Complete Step-by-step answer:
Given that
Radius of 1st circle \[{r_1} = 8{\text{ cm}}\]
Radius of 2nd circle \[{r_2} = 6{\text{ cm}}\]
The diagram of the first and second circle is as shown in the below figure:
We know that the area of the circle of radius \[r{\text{ cm}}\] is given by \[\pi {r^2}{\text{ c}}{{\text{m}}^2}\]
So, the area of 1st circle \[ = \pi {r_1}^2{\text{ c}}{{\text{m}}^2} = \pi {\left( 8 \right)^2}{\text{ c}}{{\text{m}}^2} = 64\pi {\text{ c}}{{\text{m}}^2}\]
And the area of 2nd circle \[ = \pi {r_2}^2{\text{ c}}{{\text{m}}^2} = \pi {\left( 6 \right)^2}{\text{ c}}{{\text{m}}^2} = 36\pi {\text{ c}}{{\text{m}}^2}\]
Let \[r\] be the radius of required circle as shown in the below figure:
Given that,
Area of the required circle i.e., \[\pi {r^2}{\text{ c}}{{\text{m}}^2}\] = Area of the 1st circle + Area of the 2nd circle
So, we have
\[
\pi {r^2}{\text{ c}}{{\text{m}}^2} = 64\pi {\text{ c}}{{\text{m}}^2} + 36\pi {\text{ c}}{{\text{m}}^2} \\
\pi {r^2}{\text{ c}}{{\text{m}}^2} = \left( {64 + 36} \right)\pi {\text{ c}}{{\text{m}}^2} \\
{r^2} = 100 \\
\therefore r = 10{\text{ cm}} \\
\]
Thus, the radius of the required circle is 10 cm.
Note: The area of the circle of radius \[r{\text{ cm}}\] is given by \[\pi {r^2}{\text{ c}}{{\text{m}}^2}\]. Here we have not considered the negative of \[r\] as the radius of the circle is always positive. The area of the required circle is greater than the individual areas of the given circle.
