Question

The radii of two circles are in the ratio 3:8 if the difference between their areas is $ 2695\pi $ sq. cm, find the area of the smaller circle.
A) $ 1386 $ sq. cm
B) \[1280\] sq. cm
C) \[1187\] sq. cm
D) \[1546\] sq. cm

Answer 1

Hint: We will first take a constant x and consider the area in the given ratio. Then we will take their difference as $ 2695\pi $ sq. cm then we will find the value of unknown taken and after that, the area will be found out.
Formula used:
The area of a circle of radius ‘r’ is $ \pi {r^2} $ .

Complete step-by-step answer:
We have been given that radii of two circles are in the ratio 3:8 and the difference between their areas is $ 2695\pi $ sq. cm, and we are asked to find the area of the smaller circle respectively. So let's take $ {r_1} $ as radius of one circle and $ {r_2} $ as radius of another circle.. So as both the radii of circles are in a ratio of 3:8 so we have $ \dfrac{{{r_1}}}{{{r_2}}} = \dfrac{3}{8} $ .
Now we will cross multiply them $ {r_2} = \dfrac{8}{3}{r_1} $ . We have the formula for the area of a circle with radius r that is given by $ \pi {r^2} $, So let's find the area of the individual circle. Let’s take radius as $ {r_1} $ so area will be $ \pi {r_1}^2 $, Now let's calculate the area for the second circle that is with a radius of $ {r_2} $ So the area will be $ \pi {r_2}^2 $
Which is equal to $ \pi {\left( {\dfrac{8}{3}{r_1}} \right)^2} = \dfrac{{64}}{9}\pi {r_1}^2 $
We are given the question that their difference is $ 2695\pi $ sq.
So let's write an equation satisfying the above statement: $ \dfrac{{64}}{9}\pi {r_1}^2 - \pi {r_1}^2 = 2695\pi $
Now we have $ \dfrac{{64}}{9}\pi {r_1}^2 - \pi {r_1}^2 = 2695\pi $
\[ \Rightarrow \dfrac{{64}}{9}{r_1}^2 - {r_1}^2 = 2695\]
 $ \Rightarrow \dfrac{{64 - 9}}{9}{r_1}^2 = 2695 $
 $ \Rightarrow \dfrac{{55}}{9}{r_1}^2 = 2695 $
 $ \Rightarrow {r_1}^2 = 2695 \times \dfrac{9}{{55}} $
 $ \Rightarrow {r_1}^2 = \dfrac{{24225}}{{55}} $
 $ \Rightarrow {r_1}^2 = 441 $
\[ \Rightarrow {r_1} = 21\]
Therefore ,
Area of smaller circle \[\pi {r_1}^2 = \pi \times {(21)^2} = \dfrac{{22}}{7} \times 441 = 1386\] sq.cm
So if the radii of two circles are in the ratio 3:8 if the difference between their areas is $ 2695\pi $ sq. cm , the area of the smaller circle is \[1386\] sq. cm .
So, the correct answer is “ \[1386\] sq. cm”.

Note: The circle with a smaller radius will have a smaller area and the larger radius will have more area. So while calculating the difference we will subtract the smaller area from the larger area or just we can put the modulus of the subtraction which will adjust the required thing.