Question

7. The sum of the radii of two circles is 7 cm, and the difference of their
circumferences is 8 cm. Find the circumferences of the circles.
​

Answer 1

SOLUTION :-

Let,

Radius of the first circle = $\sf r_1$

Radius of the second circle = $\sf r_2$

According to the first condition,

$\longrightarrow\sf r_1+r_2= 7 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ....................(1)$

According to the second condition,

$\longrightarrow \sf 2\pi r_1-2\pi r_2= 8 \\\\\longrightarrow 2\pi(r_1-r_2)=8 \\\\\longrightarrow r_1-r_2= \dfrac{8}{2\pi}\\\\\longrightarrow r_1-r_2 = \dfrac{8\times7}{2\times 22}\\\\\longrightarrow r_1-r_2 = \dfrac{28}{22} \\\\\longrightarrow\sf r_1-r_2 = \dfrac{14}{11}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ...................(2)$

Add equation (1) and equation (2),

$\longrightarrow\sf 2r_1=7+\dfrac{14}{11} \\\\\longrightarrow 2r_1 = \dfrac{(7\times 11)+14}{11} \\\\\longrightarrow 2r_1 = \dfrac{77+14}{11} \\\\\longrightarrow 2r_1=\dfrac{91}{11} \\\\\longrightarrow\bf r_1= \dfrac{91}{22}$

Substitute the value of $\sf r_1$ in equation (1),

$\longrightarrow \sf \dfrac{91}{22}+r_2= 7 \\\\\longrightarrow r_2= 7 -\dfrac{91}{22}\\\\\longrightarrow r_2 = \dfrac{(22\times 7 )-91}{22} \\\\\longrightarrow r_2 = \dfrac{154-91}{22}\\\\\longrightarrow \bf r_2=\dfrac{63}{22}$                            

Circumference of the first circle = $\sf 2 \pi r_1$

                                                     = $\sf 2 \times\dfrac{22}{7}\times \dfrac{91}{22}$

                                                     = $\sf \dfrac{2\times 91}{7}$

                                                      = $\sf \dfrac{182}{7}$

                                                      = $\bf 26 \ cm$

Circumference of the second circle = $\sf 2\pi r_2$

                                                           = $\sf 2 \times \dfrac{22}{7}\times \dfrac{63}{22}$

                                                           =  $\sf \dfrac{63\times 2}{7}$

                                                           =  $\sf\dfrac{126}{7}$

                                                           = $\bf 18 \ cm$