Question

the radii of two circles are 48 and 13cm. Find tthe area of the circle whcch has its circumference equal to the difference of the circumferences of the given two circles​

Answer 1

$\underline{\purple{\sf \ \ \ \ \ \ Given :- \ \ \ \ \ \ \ }}$

• Radii of two circles = 48cm and 13cm

$\underline{\sf \ \ \ \star\ To\ Find :- \ \ \ \ \ \ \ }$

• We have to find out the Area of that circle whose circumference is equal to the Difference of the circumference of given two circles

$\underline{\sf \ \ \ \star\ Solution :- \ \ \ \ \ \ \ }$

• Find the Circumference of the two circles

$\underline{\boxed{\sf{\dag\ \ Circumference\ of \ circle = 2 \pi r}}}$

• Find the circumference of circle whose radius is 48cm

$\dashrightarrow\sf Circumference\ of \ Circle_1= 2\times \dfrac{22}{7}\times 48\\ \\ \\ \dashrightarrow\sf Circumference \ of \ Circle_1={\underline{\boxed{\purple{\sf \dfrac{2112}{7}}}}}$

• Find the circumference of circle whose radius is 13cm

$\dashrightarrow\sf Circumference\ of \ Circle_2= 2\times \dfrac{22}{7}\times 13\\ \\ \\ \dashrightarrow\sf Circumference \ of \ Circle_2={\underline{\boxed{\red{\sf \dfrac{572}{7}}}}}$

• Now find out the circumference of new circle which is equal to the Difference of $\sf C_1 - C_2$

$\underline{\sf{\maltese \ Circumference\ of \ new \ circle= C_1- C_2}}$

$:\implies\sf Circumference\ of \ new \ circle = \bigg[ \dfrac{2112}{7}\bigg]- \bigg[\dfrac{572}{7}\bigg]\\ \\ \\ :\implies\sf C.\ of \ new \ circle = \cancel{\dfrac{1540}{7}}\\ \\ \\ :\implies\sf C.\ of \ new \ circle = {\underline{\boxed{\purple{\sf 220cm}}}}$

$\rule{300}{1.5}$

Now we have to find the Area of new circle

• Find out the radius !

$\dashrightarrow\sf Circumference\ of \ circle= 2 \pi r\\ \\ \\ \dashrightarrow\sf 220= 2\times \dfrac{22}{7}\times r \\ \\ \\\dashrightarrow\sf r= \dfrac{\cancel{220}\times 7}{\cancel{44}}\\ \\ \\ \dashrightarrow\sf r= 5\times 7\\ \\ \\\dashrightarrow{\underline{\boxed{\sf{\blue{ radius= 35cm}}}}}$

• Now find the Area of new circle

$\underline{\boxed{\sf{\ Area\ of \ circle= \pi r^2 }}}$

$\dashrightarrow\sf Area \ of \ circle= \dfrac{22}{\cancel{7}}\times \cancel{35}\times 35\\ \\ \\ \dashrightarrow\sf Area\ of \ circle = 22\times 5\times 35\\ \\ \\ \dashrightarrow\sf Area_{circle}= {\underline{\boxed{\sf{\purple{3850 cm^2}}}}}$

$\underline{\underline{\textsf{ Area \ of \ new \ circle = {\textbf{3850sq.cm}}}}}$

$\rule{300}{1}$

$\underline{\sf{\bigstar\ Alternate\ Method \ To \ find \ Radius \ of \ new \ circle }}$

$\dashrightarrow\sf C_1- C_2= C_{new}\\ \\ \\ \dashrightarrow\sf 2\pi r_1-2\pi r_2= 2\pi r\\ \\ \\ \dashrightarrow\sf 2\pi(r_1-r_2)= 2\pi r\\ \\ \\ \dashrightarrow\sf \cancel{2 \pi}(48-13)= \cancel{2 \pi } r\ \ \ \ \Big[\therefore\ r_1=48\ ; \ r_2= 13 \Big]\\ \\ \\ \dashrightarrow{\boxed{\sf 35= r}}$

★By using this You can easily find the area of the new circle !

Answer 2

$\mathcal{\gray{\underbrace{\blue{GIVEN:-}}}}$

• The radius of two circles are 48cm and 13cm .

Let,

✍️ The first circle be $\rm\red{C_1}$ .

✍️ And the second circle be $\rm\red{C_2}$ .

So,

• $\rm{radius\:of\:C_1\:\red{(r_1)}\:=\:48cm\:}$

• $\rm{radius\:of\:C_2\:\red{(r_2)}\:=\:13cm\:}$

$\mathcal{\gray{\underbrace{\blue{TO\:FIND:-}}}}$

• The area of a new circle, which has its circumference equal to the difference between the circumference of $\rm\red{C_1}$ to the circumference of $\rm\red{C_2}$ .

$\mathcal{\gray{\underbrace{\blue{SOLUTION:-}}}}$

$\pink\bigstar\:\rm{\green{\boxed{\orange{Circumference\:of\:Circle\:=\:2\:\pi\:r\:}}}}$

$\pink\bigstar\:\rm{\green{\boxed{\orange{Area\:of\:Circle\:=\:\pi\:r^2\:}}}}$

Circumference of first circle = $\rm\green{2\pi\:r_1}$

$\rm{=\:2\pi\times{48}\:}$

$\rm{\purple{=\:96\pi\:cm}}$

Circumference of second circle = $\rm\green{2\pi\:r_2}$

$\rm{=\:2\pi\times{13}\:}$

$\rm{\purple{=\:26\pi\:cm}}$

Let,

✍️ “r” be the radius of the new circle .

So,

$\checkmark\:\rm\green{Circumference\:of\:new\:circle\:=\:2\:\pi\:r}$

According to the question,

✔️ $\rm{2\:\pi\:r\:=\:96\pi\:-\:26\pi}$

$\rm{\implies\:2\:\pi\:r\:=\:70\pi}$

$\rm{\implies\:r\:=\:\dfrac{70\pi}{2\pi}\:}$

$\rm{\purple{\implies\:r\:=\:35cm\:}}$

Hence,

✍️ Area of the new circle = $\rm\green{\pi\:r^2}$

$\rm{\implies\:Area\:of\:the\:new\:circle\:=\:\dfrac{22}{7}\times{(35)^2}\:}$

$\rm{\implies\:Area\:of\:the\:new\:circle\:=\:\dfrac{22}{7}\times{35}\times{35}\:}$

$\rm{\implies\:Area\:of\:the\:new\:circle\:=\:22\times{5}\times{35}\:}$

$\rm{\purple{\implies\:Area\:of\:the\:new\:circle\:=\:3850cm^2\:}}$

$\rm\red{\therefore}$$<font color=baby>$ The area of the new circle is 3850cm².