Question

The radii of two circles are 8cm and 6cm respectively. Find the radius of the circle having its area equal to the sum of the areas of the two circles.
(a) 32cm
(b) 23cm
(c) 15cm
(d) 10cm

Answer 1

Hint: First we will find the areas of the two circles with radii 8cm and 6cm respectively using the formula $A=\pi {{r}^{2}}$. Then we will add the areas to find the area of the new circle which will help us to find the radius of such a circle.

Complete step-by-step solution:
Let \[{{r}_{1}}\] and ${{r}_{2}}$ represent the radii of the two circles and \[{{A}_{1}}\] and \[{{A}_{2}}\] represent the area of those circles.
Let ${{r}_{3}}$ and \[{{A}_{3}}\] be the radius and area of the circle formed by summing the areas \[{{A}_{1}}\] and\[{{A}_{2}}\].
According to the question,
\[{{r}_{1}}=8cm\]
\[{{r}_{2}}=6cm\]
We know that the area \[A\] of a circle of a radius \[r\]is given by
\[A=\pi {{r}^{2}}\text{}\ldots \left( i \right)\]
Then applying this formula to find the areas \[{{A}_{1}}\] and \[{{A}_{2}}\], we get
 \[\begin{align}
  & {{A}_{1}}=\pi {{r}_{1}}^{2}=\pi {{\left( 8 \right)}^{2}}=\pi \cdot 64=64\pi c{{m}^{2}} \\
 & {{A}_{2}}=\pi {{r}_{2}}^{2}=\pi {{\left( 6 \right)}^{2}}=\pi \cdot 36=36\pi c{{m}^{2}} \\
\end{align}\]
The area \[{{A}_{3}}\] of the circle formed by adding the areas \[{{A}_{1}}\] and \[{{A}_{2}}\] is given by
\[\begin{align}
  & {{A}_{3}}={{A}_{1}}+{{A}_{2}} \\
 & =64\pi +36\pi \\
 & {{A}_{3}}=100\pi c{{m}^{2}}
\end{align}\]
Putting this value for the new circle formed in equation$\left( i \right)$ , we get
$\begin{align}
  & {{A}_{3}}=\pi {{r}_{3}}^{2}=100\pi \\
 & r_{3}^{2}=100 \\
 & r_{3}^{2}={{\left( 10 \right)}^{2}} \\
 & {{r}_{3}}=10cm \\
\end{align}$
Hence, the correct answer is an option (d).

Note: We could have put the value of $\pi $ as 3.14 or $\dfrac{22}{7}$ while calculating the areas \[{{A}_{1}}\] and\[{{A}_{2}}\], and then again while calculating the radius ${{r}_{3}}$. But it can lead to errors in decimals. So try to avoid unnecessary calculations and while doing them, always be careful and double-check.