Question

The difference of areas of two circle is 100cm^2 and the difference of their perimeter is 10cm find of their radius

Answer 1

Answer:

The radius of larger circle is 475 / 44 cm and radius of smaller circle is 405 / 44 cm.

Step-by-step explanation:

It is given that the difference of areas of two circle is 100cm^2 and the difference of their perimeter is 10cm.

Let a large circle with the radius of "a" and the radius of the smaller circle be "b",

As given in the question,

⇒ Difference of areas = 100 cm^2

⇒ Area of large circle - Area of small circle = 100 cm^2

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From the properties of circle, we know : -

Area of circle = πr^2, where r is the radius of the circle

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= > πa^2 - πb^2 = 100 cm^2

= > π( a^2 - b^2 ) = 100 cm^2    ...( $\it{i}$ )

Also,

⇒ Difference of their perimeter = 10 cm

⇒ Perimeter of large circle - perimeter of small circle = 10 cm

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From the properties of circle, we know : -

Perimeter of circle = 2πr

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= >  2πa - 2πb = 10 cm

= >  2π( a - b ) = 10 cm         ...( $\it{i}$)

Now,

= >  π( a^2 - b^2 ) = 100 cm^2   ---: ( 1 )

= >  2π( a - b ) = 10 cm               ---: ( 2 )

Divide ( 1 ) by ( 2 ) : -

$\implies \dfrac{\pi (a^2-b^2)}{2\pi ( a - b ) }=\dfrac{100\:\text{cm^2}}{10\:\text{cm}}\\\\\\\implies \dfrac{(a-b)(a+b)}{2(a-b)}=10\text{cm}\\\\\\\implies \dfrac{a+b}{2}=10\text{cm}\\\\\\\implies a + b= 20\text{cm}$

Now,

= > 2π( a - b ) = 10 cm

= > a - b = 10 cm x 1 / 2 x 7 / 22

= > a - b = 35 / 22 cm

Then, adding a + b and a - b,

a + b = 20 cm

a - b = 35 / 22 cm        

2a    = 20 + 35 / 22 cm

= >  2a = ( 440 + 35 ) / 22 cm

= >  2a = 475 / 22 cm

= >  a = 475 / 44 cm

And, a + b = 20 cm

        b = 20 cm - 475 / 44 cm

        b = 405 / 44 cm

Hence, radius of larger circle is 475 / 44 cm and radius of smaller circle is 405 / 44 cm.

Answer 2

Answer:

475/44 cm, 405/44 cm

Step-by-step explanation:

Let the radius be r₁ and r₂

(i) Difference of areas:

Area of circle 1 (r₁) = πr₁²

Area of circle 2(r₂) = πr₂²

Difference of areas of two circle is 100 cm².

πr₁² - πr₂² = 100

π(r₁² - r₂²) = 100

(ii) Perimeter:

Perimeter of 1st circle = 2πr₁

Perimeter of 2nd circle = 2πr₂

Difference of Perimeter is 10 cm

2πr₁ - 2πr₂ = 10

2π(r₁ - r₂) = 10

π(r₁ - r₂) = 5

On solving (i) & (ii), we get

⇒ π(r₁² - r₂²)/π(r₁ - r₂) = 100/5

⇒ π(r₁ + r₂)(r₁ - r₂)/π(r₁ - r₂) = 20

⇒ r₁ + r₂ = 20            --------------------------- (iii)

Now, Equation (ii) can be written as,

π(r₁ - r₂) = 5

(r₁ - r₂) = 35/22

On solving both the equations, we get

r₁ + r₂ = 20

r₁ - r₂ = 35/22

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2r₁ = 475/22

r₁ = 475/44

Substitute r₁ = 475/44 in (iii), we get

r₁ + r₂ = 20

(475/44) + r₂ = 20

r₂ = 20 - (475/44)

r₂ = (880 - 475)/44

r₂ = 405/44

Therefore, the radius r₁ = 475/44 cm and r₂ = 405/44 cm.

Hope it helps!