Question

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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

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:\huge\underline{\pink{\sf \ \: Given :- }}$

Radii of two circles = 48cm and 13cm

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$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \large \sf\orange {To\ Find }$

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

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$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \large \sf \blue { Solution :- }$

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

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

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Find the circumference of circle whose radius is 13cm

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

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

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

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

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Now we have to find the Area of new circle

Find out the radius !

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

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Now find the Area of new circle

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

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

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

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

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

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