Question

The sum of the radii of the two circles is 14 cm and the difference of their circumference is 15 cm. Find the radii of the two circles.
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Answer 1

Answer:

The radii of the two circles are 5.81 cm & 8.19 cm.

Step-by-step-explanation:

Let the radius of the greater circle be R.

And the radius of the smaller circle be r.

Let the circumference of the greater circle be C.

And the circumference of the smaller circle be c.

Now, from the first condition,

Sum of the radii is 14 cm.

∴ R + r = 14

⇒ R = 14 - r - - ( 1 )

Now, we know that,

Circumference of circle = 2 π r

∴ C = 2 π R

⇒ C = 2 π ( 14 - r ) - - [ From ( 1 ) ]

⇒ C = 28 π - 2 π r

Now,

c = 2 π r

Now, from the second condition,

The difference between the circumferences of the circles is 15 cm.

∴ C - c = 15

⇒ ( 28 π - 2 π r ) - ( 2 π r ) = 15

⇒ 28 π - 2 π r - 2 π r = 15

⇒ 28 π - 4 π r = 15

⇒ 4 π ( 7 - r ) = 15

⇒ ( 7 - r ) = 15 ÷ 4 π

⇒ 7 - r = 15 ÷ 4 * 3.14

⇒ 7 - r = 15 ÷ 12.56

⇒ 7 - r = 1.19

⇒ r = 7 - 1.19

⇒ r = 5.81

Now, by substituting r = 5.81 in equation ( 1 ), we get,

R = 14 - r - - ( 1 )

⇒ R = 14 - 5.81

⇒ R = 8.19

∴ The radii of the two circles are 5.81 cm & 8.19 cm.

Answer 2

Answer :-

$\: \: \underline{\rm{\green{\mapsto \: \: Let's \: understand \: the \: concept}}}$

Here we see that we are given two unknown values. Now if we need to find both of them, the simplest way is using Linear Equations in Two Variables, using which we can solve this question. Let's do it !!

• Circumference of circle = 2πr

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★ Question :-

The sum of the radii of the two circles is 14 cm and the difference of their circumference is 15 cm. Find the radii of the two circles.

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★ Solution :-

Given,

» Sum of radii of two circles = 14 cm

» Difference in the circumference of circles = 15 cm

• Let the radius of smaller circle be 'x' cm

• Let the radius of the larger circle be 'y' cm.

Then, according to the question :-

~ Case I :-

⌬ y + x = 14 cm

⌬ y = 14 - x ... (i)

~ Case II :-

We know that, the circumference of the circle with larger radius will be more than the other. Then,

⌬ 2πy - 2πx = 15 cm

⌬ 2π(y - x) = 15 ... (ii)

From equation (i) and (ii), we get,

⌬ 2π(14 - x - x) = 15

⌬ 2π(14 - 2x) = 15

* Here we are taking the value of π = (22/7) .

$\: \: \huge{\purple{\Longrightarrow \: \: \: (2 \: \times \: \dfrac{22}{7} \: \times \: 14) \: - \: (2 \: \times \: \dfrac{22}{7} \: \times \: 2x) \: = \: 15}}$

⌬ (2 × 3.14× 14) - (2 × 3.14 × 2x) = 15

⌬ 87.92 - 12.56x = 15

⌬ -12.56x = 15 - 87.92

⌬ -12.56x = -72. 92

$\: \: \huge{\rm{\orange{\longrightarrow \: \: x \: = \: \dfrac{\not{-}72.92}{\not{-}12.56}}}}$

⌬ x = 5.81 cm

• So the radius of smaller circle = x = 5.81 cm

• Hence, the radius of larger circle = y = 14 - x = 8.19 cm

$\: \: \underline{\boxed{\rm{\red{\mapsto \: \: \: Thus, \: the \: radius \: of \: larger \: circle \: is \: \underline{\blue{8.19 \: cm}} \: and \: that \: of \: smaller \: circle \: is \: \underline{\blue{5.81 \: cm}}}}}}$

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$\: \: \underline{\boxed{\bf{\pink{\mapsto \: \: Confused? \: Don't \: worry \: let's \: verify \: it}}}}$

For verification, we need to simply apply the values we got into the equations we formed. Then,

~ Case I :-

=> y = 14 - x

=> 8.19 = 14 - 5.81

=> 8.19 = 8.19

Clearly, LHS = RHS.

~ Case II :-

=> 2πy - 2πx = 15 cm

=> (2 × 3.14 × 8.19) - (2 × 3.14 × 5.81) = 15

=> 51.496 - 36.496 = 15

=> 15 = 15

Clearly, LHS = RHS

Here both the conditions satisfy, so our answer is correct. Hence, Verified.

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$\: \: \underline{\boxed{\sf{\blue{Reference \: for \: Supplementary \: Counsel \: :-}}}}$

• Area of Circle = πr²

• Area of Rectangle = Length × Breadth

• Area of Square = (Side)²

• Area of Triangle = ½ × Base × Height

• Perimeter of Rectangle = 2 × (Length + Breadth)

• Perimeter of Square = 4 × Side