Question

If the area of circle is equal to sum of the areas of two circles of diameters 10cm and 24cm, then the diameter of the larger circle (in cm) is:
(a) 34
(b) 26
(c) 17
(d) 14

Answer 1

Hint: We will first find the radius of both circles having diameter 10cm and 24cm using the formula $ radius=\dfrac{diameter}{2} $ . We will assume the radius of the larger circle to be R and $ {{r}_{1}} $ , $ {{r}_{2}} $ for diameter 10cm , 24cm respectively. Then we will use the formula of area of the circle given as $ Area=\pi {{r}^{2}} $ . As per the statement given, we will have equation as $ \pi {{R}^{2}}=\pi {{r}_{1}}^{2}+\pi {{r}_{2}}^{2} $ . On solving this and putting values we will get a value of R. Then again, we will use the formula $ radius=\dfrac{diameter}{2} $ we will get diameter value.

Complete step-by-step answer:
Here, we are given that the area of the circle is equal to the sum of areas of two circles having a diameter of 10cm and 24cm. So, first we will draw the figure for this.

Now, we will consider radius of larger circle to be R. Radius of circle with diameter 10cm will be
 $ {{r}_{1}}=\dfrac{diameter}{2}=\dfrac{10}{2}=5cm $ and radius of circle with diameter 24cm will be equal to $ {{r}_{2}}=\dfrac{diameter}{2}=\dfrac{24}{2}=12cm $ .
Now, we will write the mathematical form of the statement that the area of a circle is equal to the sum of areas of two circles having diameters of 10cm and 24cm. We know that area of circle is given by using the formula $ Area=\pi {{r}^{2}} $
So, we get as
 $ \pi {{R}^{2}}=\pi {{r}_{1}}^{2}+\pi {{r}_{2}}^{2} $ ………………..(1)
Now, we will take $ \pi $ common from both sides and after cancelling that term we will get as
 $ {{R}^{2}}={{r}_{1}}^{2}+{{r}_{2}}^{2} $
We will substitute the radius value we have found above. So, we will get as
 $ {{R}^{2}}={{\left( 5 \right)}^{2}}+{{\left( 12 \right)}^{2}} $
On further solving, we get as
 $ {{R}^{2}}=25+144=169 $
Thus, on taking the square root on both sides, we will get radius of larger circle as $ R=13cm $
So, the diameter of the larger circle will be $ R=\dfrac{diameter}{2} $ . On putting values, we get as $ diameter=R\times 2=13\times 2=26cm $ .
Thus, diameter is 26cm.
So, the correct answer is “Option B”.

Note: Another approach of solving this is by substituting value of radius in form of diameter in formula of area of circle i.e. we can write $ Area=\pi {{r}^{2}} $ as $ Area=\pi {{\left( \dfrac{d}{2} \right)}^{2}} $ . So, we get equation as $ \pi {{\left( \dfrac{D}{2} \right)}^{2}}=\pi {{\left( \dfrac{{{d}_{1}}}{2} \right)}^{2}}+\pi {{\left( \dfrac{{{d}_{2}}}{2} \right)}^{2}} $ . On substituting the values and cancelling the same terms, we get as $ {{D}^{2}}={{\left( 10 \right)}^{2}}+{{\left( 24 \right)}^{2}}=676 $ . Thus, on taking the square root we will get D as 26cm.