Question

Find the radius of a circle whose area is equal to the difference of the areas of two circles of radii 6 cm and 10 cm respectively



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Answer 1

Answer :-

• The radius of the circle is 8 cm.

To find :-

• The radius of a circle whose area is equal to the difference of the areas of two circles of radii 6 cm and 10 cm respectively.

Step-by-step explanation :-

• Before finding the radius of the circle, we first have to find it's area and for that, we have to find the areas of the two circles of radii 6 cm and 10 cm respectively.

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Finding the area of the first circle :-

We know that :-

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

Where,

• π = pi.
• r = Radius.

Here,

• pi = 22/7.
• Radius = 6 cm.

Hence,

$\implies\rm Area \: of \: the \: circle = \dfrac{22}{7} \times 6 \times 6$

Multiplying 6 with 6,

$\implies\rm Area \: of \: the \: circle = \dfrac{22}{7} \times 36$

Multiplying the remaining numbers,

$\implies\rm Area \: of \: the \: circle = \dfrac{792}{7}$

Divising 792 by 7,

$\implies\rm Area \: of \: the \: circle = 113.142857 \: {cm}^{2}$

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Finding the area of the second circle :-

As given above,

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

Where,

• π = pi.
• r = Radius.

Here,

• pi = 22/7.
• Radius = 10 cm.

Hence,

$\implies\rm Area \: of \: the \: circle = \dfrac{22}{7} \times 10 \times 10$

Multiplying 10 with 10,

$\implies\rm Area \: of \: the \: circle = \dfrac{22}{7} \times 100$

Multiplying the remaining numbers,

$\implies\rm Area \: of \: the \: circle = \dfrac{2200}{7}$

Dividing 2200 by 7,

$\implies\rm Area \: of \: the \: circle = 314.285714 \: cm^2$

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Now, it has been given that :-

• The area of the circle is equal to the difference of the areas of two circles of radii 6 cm and 10 cm respectively.

Difference between their areas is :-

$\rightarrow\boxed{\bf314.285714 - 113.142857}$

$\rightarrow\boxed{\bf201.142857\: {cm}^{2}}$

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• Hence, the area of the circle is 201.142857 cm². Now let's find it's radius!

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

Where,

• π = pi.
• r = Radius.

Here,

• pi = 22/7.
• Area = 201.142857 cm².

Substituting the given values in this formula,

$\longmapsto\tt201.142857 = \dfrac{22}{7} \times r^{2}$

Transposing 7 from RHS to LHS, changing it's sign,

$\longmapsto\tt201.142857 \times 7 = 22 \times {r}^{2}$

Multiplying 201.142857 with 7,

$\longmapsto\tt1408 = 22 \times {r}^{2}$

Transposing 22 from RHS to LHS, changing it's sign,

$\longmapsto\tt\dfrac{1408}{22} = {r}^{2}$

Dividing 1408 by 22,

$\longmapsto\tt64 = {r}^{2}$

Now let's find the square root of 64 to find the radius.

$\longmapsto\tt \sqrt{64} = r$

Finding the square root of 64,

$\longmapsto\overline{ \boxed{ \tt8 \: cm = r}}$

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• Hence, the radius of the circle is 8 cm.

Answer 2

$\tt\small\underline\red{Given:-}$

$\tt{\implies Radius\:_{(two\:circle)}=6cm\:and\:10cm}$

$\tt\small\underline\red{To\:Find:-}$

$\tt{\implies The\:radius\:_{(new\:circle)}=?}$

$\tt\small\underline\red{Solution:-}$

To calculate the radius of new circle at first we have to assume the radius of 1st circle be r¹ and the radius of 2nd circle be r². The radius of new circle be r. According to given question we have to set up equation. Then applying the formula of area of circle to calculate the radius of new circle. Here the radius of 1st circle be 10cm and the radius of 2nd circle be 6cm.

$\tt\small\underline\red{Formula\:used:-}$

$\tt{\implies Area\:of\:circle=\pi\:r^2}$

$\tt Difference\:_{(area\:of\:2\:circle)}$

$\tt{\implies \pi\:(r^1)^2-\pi\:(r^2)^2=\pi\:r^2}$

$\tt{\implies \pi\:(10)^2-\pi\:(6)^2=\pi\:r^2}$

$\tt{\implies \pi(10^2-6^2)=\pi\:r^2}$

$\tt{\implies \pi(100-36)=\pi\:r^2}$

$\tt{\implies \pi(64)=\pi\:r^2}$

$\tt{\implies r^2=64}$

$\tt{\implies r=8cm}$

$\tt\large{Hence'}$

$\tt The\:radius\:new\:circle$

$\sf\small\underline\red{Some\: important\: Formula:-}$

Area of The Circle-:

$\pi\:r^2$

$\tt{\implies Perimeter\:_{(semicircle)}=\pi\:r}$

$\tt Area (semicircle)=π\:r^2\;2$

$\tt{\implies Area\:_{(sector)}=\dfrac{\theta}{360}\times\:\pi\:r^2}$

$\tt{\implies length\:_{(circle\:arc)}=\dfrac{\theta}{360}\times\:2\pi\:r}$

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